Can anyone give me a grammatical explanation as to why "that being said" is proper English?

That being said, ... can be rewritten as After saying all that, ... .

In the statement said is acting as an adjective, so is equivalent to that statement being said, or analogous to that ball being bouncy. You could point at a ball and say that being bouncy, you can use it to entertain your dog.

It is fine to use said in an adjectival form in this sense, because it is describing the state of a statement; the state of having been spoken.

Said being used as an adjective is not unusual, even if it is chiefly used in law: http://dictionary.reference.com/browse/said - although this is a different type of usage to how said is used in "that being said".

That being said, your acquaintance is being a grammar maven, so can be ignored. ;)

"That being said" I interpret as "That, now in a state of in the record of the current conversation, ..."

So it's similar to "That being a Ford..." or "That being green". "Said" is an adjective in this case.

This is easily refuted in this way:

You, sir, take a prescriptive view of grammar. "That being said" is common usage, unambiguous, and you should adapt your understanding of English grammar to include this construction.

The same response should be applied to any situation when someone is making grammatical prescriptions which are not related to correcting ambiguous or mis-communicative expressions.

Your pedant is completely wrong, not just because he's protesting in futility to a well-established idiom, but because his grammatical analysis of the construction is mistaken.

That being said is an adverbial participle phrase. Note that the verbal portion of the phrase, being said, does not contain a finite verb, and only finite verbs are tensed. (This is not just a fact of English, but a general facet of Western European verbal systems: non-finite verb forms generally indicate aspect and voice, but not tense.) The terminology is somewhat confused on this point since being is often called the "present participle", but there is in fact nothing that specifies the present tense about this construction, as the following examples show:

The lawn was being mowed yesterday.

The aquarium will be being cleaned all day tomorrow.

In all of these cases, the bolded portion does not contribute any tense information at all -- it rather indicates the progressive aspect and passive voice. The tense is entirely indicated by the finite verb in italics.

The idiomatic phrase that being said contains no finite verb and no tense. Your pedant has merely demonstrated his own ignorance of English grammar. There is absolutely no reason to object to this idiom.

You can use "That being dealt with" the same way, and it sounds O.K. to my ear. Google finds a number of examples of this, for instance:

That being dealt with, Prospero now goes to meet the shipwrecked King & Co.

Of course, your friend can argue that these are all incorrect, as well.

"That being said" is perfectly fine logico-grammatically, as others have explained. Side step the issue and say

That said,

It's shorter and says the same thing.

Short answer: It is fine.

Lets begin with some examples utilising the same structure.

"It is said that he could lift an elephant with one hand."

"It is believed that these are the first cave drawing to have ever been done."

In these examples, 'said' and 'believed' are adjectives formed from the participle.

Consider: a pair of ripped jeans -or- "an old moth-eaten suit"

In both cases, the adjectives preceding the noun are formed from past participles. They could also be written like this:

The jeans had been ripped and thus they looked terrible.

The suit had been eaten by moths so he threw it out.

The form enquired about uses this same construction and is therefore perfectly acceptable:

It is said that he could lift an elephant with one hand.

The locals believe he could lift an elephant with one hand. That being said, they also believe the moon is made of cheese.

Hope that helps.

I think your friend has a point. I will not comment on whether this construction is acceptable; I will just explain why it is somewhat irregular. Consider this sentence:

Its walls being assaulted by the Crusaders, Constantinople got little sleep that night.

It is clear that the city's being assaulted does not precede the waking of its inhabitants: being assaulted happens at the same time as got little sleep. That is how present participles work: they happen at the same time as the main verb in the sentence of which they are a part. The absolute construction with being + past participle inherits (as JSBangs said) the time of got.

But assaulted is a past participle, you say: its action should precede the main verb in time! That is generally true, as in the following example:

The captured baron paid a ransom of two thousand marks.

Here the baron's being captured precedes his paying a ransom: he is not still in the process of being captured as he pays the ransom. But that does not apply to the Constantinople example. The reason for this is that the auxiliary verb to be changes the meaning of the past participle: usually, past participle + to be loses the sense of being earlier in time that is normally present in the past participle.

Back to the said example. You didn't provide a context, but based on your comparison with having been said I believe you were thinking of a sentence like this:

That being said, I will now proceed to a new topic: Borgia dominance in the Vatican.

If we assume that being + past participle happens at the same time as the verb of the main clause, and that I will proceed begins now, the "saying" of that being said should still be going on now. But I don't think this is what it is supposed to mean. That being said is here most probably used as a substitute for that said and that having been said: the speaker wants to tell us that the previous topic is finished. Then the phrase should be considered irregular.

---

However, perhaps we could take said as an adjective, not a past participle. Whenever a participle is used as an adjective, which means that it is not linked to an auxiliary verb like to be, it still has its sense of being earlier than the main verb, as in the captured baron above. But is this possible? There are some examples of this with other past participles:

Caesar, this catapult must be broken.

You could read this as Caesar, we must arrange for this catapult to be broken: then be broken is treated as auxiliary be + past participle. Or it could be Caesar, this catapult must be defective (how else do you explain why it failed to fire?): then broken is treated as an adjective and keeps its sense of having happened earlier; the catapult was broken some time ago and as a result it is now in a broken state. Note that be is a copula then, not an auxiliary verb.

Can we so interpret said in that being said? Then I'd paraphrase it as this being in a said state, I will now proceed. That is possible, but it certainly isn't the usual way to interpret said. Whereas broken is very frequently used as an adjective, probably much more often than as a true participle, this is not the case with said. Even so, I think this may very well be how those who use this construction subconsciously do it.

However, though this construction is quite common, I do not think it has any advantage over that said, which is both shorter and less convoluted. As far as style is concerned, I prefer that said.

---

If that being said is used to express an argument that has already been presented ("said") but is still being examined by the speaker, its sense of happening at the same time isn't a problem at all:

Most modern scholars say that Churchill was one of the few who had no sympathy for Hitler at all. That being said, it is no wonder that Chamberlain happily accepted Hitler's guarantee that he'd leave Poland alone.

Here that being said means considering that they say this (notice the present simple); the scholars never really stop saying this as long as we keep reading what they have written or using their arguments. This is similar to the timeless it says "he was a great poet" on his grave: it says so as long as the letters are still readable. This usage is not irregular at all. However, I don't think this is what the question was about, because it compared that being said to that having been said and that said, which indisputably have a sense of being earlier than the main verb. Substituting that having been said in the Churchill example would change its meaning completely.

The idiom is "that said". It's a short form of "that having been said". "That being said" is a more recent corruption, and it is only acceptable because it has become so widespread that there is little choice but to accept it. However, it does not parse grammatically to my ear. It is the equivalent of "that being thrown" or "that being opened". In those instances correct usage, as I was taught a million years ago, anyway, would require "having been". The phrase in question is a pompous affectation in any event, but "at the end of the day" it so overworked that it can hardly be considered "out of the box".

Edit 2: In short, it is grammatical because said is an adjective in this context, not a verb. This is the only grammatical interpretation of the phrase.

Consider if I were to write a paragraph and follow it with "That being verbose, you probably didn't read it all." I am describing the state of the paragraph which came before. It is verbose. Similarly, I can say that it is said, which is a phrase meaning "it has been said."

Edit: "Being said" in this context does not mean "being said right now." It means "being already said." "Said" is used as an adjective, not a verb. Consider the sentence "It's too late; the words are said."