Avis: d4, d4, d10, d12, dX (where X is d4, d6, d8, d10, d12, or d20)
Iago: d20, d20, d20, dX (where X is d4, d6, d8, d10, d12, or d20)
Note: Technically speaking according to the Button Men rules, the dX swing die can be any number within the range of 4 to 20 inclusive. For the purposes of this question, we will assume that the swing die will be one of the polyhedral dice listed.
If you want to take a stab at this question, be my guest. I will be attempting to brute force an answer over the next couple of days. If you want to post whether you believe brute forcing an answer is possible, and the total number of permutations that need to be examined, that would be nice (I will attempt to make a guess myself).