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From the point of view of what you call predicative set theory --- I would say "predicativism given the natural numbers" --- I don't think there are any known arguments for the consistency of ZF, and such a thing seems very unlikely. The proof-theoretic strength of natural predicative theories are quite weak, generally around the level of PA. You can push this up a bit, but in order to show that ZF is consistent you would need proof-theoretic ordinals vastly beyond anything that anyone thinks is predicative.

(The generally accepted ordinal limit of predicative theories is $\Gamma_0$, but this is incorrect. I do not say "I believe" or "it seems": the analysis that concludes $\Gamma_0$ is hopelessly wrong. I explain why in this paper.)

I realize that you are only asking for an "informal" argument, but I don't see how that really helps. Looking at the question from the point of view of proof-theoretic ordinals, I think it's clear that ZF is utterly out of reach.

The best a predicativist can do with ZF may just be to assign some credence to its consistency based on the fact that no inconsistency has been found yet. I'm not sure how strongly that evidence should be weighed. It's also true that the consistency of ZF is implied by various, arguably natural arithmetical statements; Harvey Friedman is known for his work on this. Possibly that could be considered more reason to believe consistency.

I want to emphasize, though, that people often talk about consistency as if that is the only thing that matters. Surely, if you are a predicativist, you should care not only about whether ZF is consistent, but also about whether it proves true arithmetical theorems. You want it to be arithmetically sound, not just consistent. For instance, if ZF proves that Turing machine $x$ halts on null input, for some specific value of $x$, we should care about whether this is actually the case. It could well be consistent while proving false statements of this type. I made this point here.

EDIT: in the comments, Ingo Blechschmidt suggests that "(apparent) consistency of impredicative systems is an unexplainable mystery from a predicative point of view". I'd say this is less of a mystery than it seems, when you remember that there have been many formal systems for various types of impredicative mathematics over the years which did turn out to be inconsistent. Most notably, the very first, Frege's Grundgesetze.

So instead of saying "Wow, all these formal systems for impredicative mathematics turned out to be consistent, isn't that amazing!" we should say "Wow, all these formal systems for impredicative mathematics turned out to be consistent, except for the ones that didn't. Maybe not so amazing."