A question on grammar in relation to pronouns

But he insisted on going, him/he being a tenacious problem solver.

Both versions are incorrect and non-idiomatic. (As an aside “tenacious problem solver” sounds like a calque from another language.)

The intended meaning is:

But he insisted on going, as he was/is/claimed to be, etc., a tenacious problem solver.

It should be

“But he insisted on going, his being a tenacious problem solver.” Or “Being a tenacious problem solver, he insisted on going.

"[his] being a tenacious problem solver" is a noun clause acting as a free modifier of “but he insisted on going.”

But he insisted on going, him/he being a tenacious problem solver.

Both forms are possible. However, the accusative is informal and somewhat unlikely, while the construction itself is relatively formal, so there's a clash of styles with the accusative tending to sound out of place here.

If you prefer "him", then go ahead and use it: I just want to point out the stylistic difference.

Regarding your second question. I'm inclined to say that "problem-solver" is a compound noun and hence should be hyphenated. Other examples with a noun + deverbal er noun structure include "factory-worker", "window-cleaner", "city-dweller", "store-manager".

One method of approaching such questions is to construct partial sentences, substituting nouns and pronouns.

John insisted on going.

He insisted on going.

No one would dispute that these are both grammatically correct. “John” and “He” are the subjects of the sentence.

John, being a tenacious problem solver, insisted on going.

Is this grammatically correct? Yes: “being a…” is in apposition to or describes John. If so,

He, being a tenacious problem solver, insisted on going.

must be correct.

Now the example sentence just involves — perhaps unnecessary, or perhaps emphatic — repetition of “John”, which in both fragments above is replaced by “he”, not “him”. Therefore, in standard English (rather than non-standard English, which may be common in certain sections of particular societies and the possible reason for the poster’s confusion) the sentence should read:

But he insisted on going, he being a tenacious problem solver.

There is no grammatical justification for changing the second (repeating) pronoun to the object/accusative case.

But, being a tenacious problem solver, he insisted on going.

The phrase 'being a tenacious problem solver' when used at the start offers clarity, and avoids confusion of agreements like "who is being?"