At the same time I knew that the final step had to involve 3 coins or less
Except that's not true.
The balanced strategy gives
1. AAAA BBBB CCCC <- 3 groups of 4 coins.
2. ABBB CCCB <- This is the next weighing if in step 1 AAAA != BBBB (otherwise it is trivial)
3. AB or BBB <- this can be resolved in one weighing AND also knowing the result of step 2.
You cannot end up with a group of 4 coins in step 3 and in one more weighing know which single coin is counterfeit, even if you know which way it leans.