Michael Dorfman stressed the following unavoidability in many answers:
Note that due to Agrippa's Trilemma, there are only three things logic could possibly be founded upon: unsupported axioms we take on faith, circular reasoning, or an infinite regress. Or, of course, a combination of the three.
I was trying to make use of the trilemma in the context of sound deduction systems for monadic second-order logic with standard semantics. Then I noticed that the presence of an (incomplete) axiom system (which I considered as given and hence didn't want to question in that context) might add further possibilities not covered by the trilemma.
Let's be clear from the start that transfinite induction needs an axiom system to work reliably, so any proof using transfinite induction might already be covered by the "unsupported axioms" part of the trilemma. My confusion/question is whether the "combination of the three" part is also correct. The "unsupported axioms" proof-method can be used both "top-down" and "bottom-up", but both "circular reasoning" and "infinite regress" look like they can only be used "top-down". Because "transfinite induction" looks like it can only be used "bottom-up", the truth of the "combination of the three" part seems to imply that any deduction system trying to use transfinite induction to overcome "incompleteness" will be forced to ultimately boil down to only "unsupported axioms", and hence will ultimately be covered by the normal "first-order" incompleteness theorems for second-order logic.
Appendix: Normal (mathematical) induction allows to capture certain cases where it is known how to systematically prove each instance in a series of successively more complex statements in a compact way. But (the simplest) transfinite induction allows to assume that all infinitely many statements that can in principle be proven true by a given (normal) induction scheme are already proven true, and hence can be used to prove a statement based on the simultaneous truth of all these "prior" statements.