How correct is the Southern colloquialism "y'all" in correct English grammar?

Yes. "Y'all" is a contraction for "you all", and so you have placed the apostrophe correctly to account for the missing letters. (You will frequently see the incorrect form "ya'll", but this doesn't expand correctly.)

Also be aware that "y'all" is always plural. Never use "y'all" to refer to a single person! You wouldn't use "you guys" or "all of you" to refer to one person, so don't use "y'all" either.

Y'all be is never grammatically correct

Y'all is never grammatically correct in UK English. It isn't used. It would be understood however, and recognised as American English.

'You' would be used instead, to represent the 'collective you' (there is no way to distinguish the collective from the individual 'you' in UK English, as far as I know. I'm a native English speaker).

Meaning: You 'The one or ones being addressed' https://www.merriam-webster.com/dictionary/you

Your question relates to a colloquial usage, perhaps a regional dialect. There are may of these. So in Somerset you will hear "Where's he to?" for "Where is he?". And if the shop assistant says "that'll be ten pounds ninety nine pence, my lover", she is not claiming previous carnal knowledge with you. The issue is not whether it is correct British or American English. It might be said to be 'non-standard'. Or so I thought. Until I checked the Cambridge English online dictionary, which says:

pronoun US informal...(you all)

'informal' has nothing to do with grammar or meaning. It is a term of disapproval. Formal English is just an approved style for communications which you want to be taken seriously.

Merriam Webster, a (perhaps 'the') US dictionary, says much the same.

pronoun ... variant of you all ... chiefly Southern ... YOU usually used in addressing two or more persons.

There is no hint here of the condescension of the word 'informal'. I think, though I cannot prove it, that it is more often used in speech than in writing other than fiction and Facebook.