In other words, for some reason they expect the universe to be more "understandable" or "sensible" than this . . .
Of course the strong AP is ludicrous in the extreme ;)
There's no number system you can invent where the fundamental constants turn into simple integers or have obvious ratios.
So why do they have those values? What gave them those values? The anthropic principle is disliked because it rejects further inquiry and progress.
Unfortunately, that doesn't explain how these values are generated. That is the important question.
No, their opposition is to the fact that a physical constant is a mere accident, a mere facet coming with the universe as a set, and not explained by some more fundamental mechanism.