Wouldn't this just become a game of picking the highest possible die that denies your opponent his highest?
IE: in a scenario where I have a 6, 4, 2, 1 and he has a 4, 4, 2, 2 and the dice available are (for argument sake) 5, 4, 3, 3, 2, 1
I would always pick the 4 first, which would then force my opponent to pick 3. Then I pick the 2 next, forcing my opponent to pick 1. etc.
Or can you combine dice, which would be a little different but I'm not sure it would actually solve anything.
If you can combine the dice the scenario would play out (in this case) exactly as "I always take the highest die".
I did have an interesting thought - have you gone back and looked at the 90s game Dragon Dice (it's actually still around). Dungeons and Dragons' dice game.... had some interesting and clever RPG-esque mechanics to it.
Kinda totally different from where you've been going though