A question on grammar in relation to pronouns

Solution 1:

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".