Is it possible to tell fasm to use multibyte NOPs for alignment instead of the usual single-byte ones? I.e. I'd like fasm to be able to choose the opcode depending on how much space it needs to occupy — without me inspecting the code and putting the needed byte sequence manually.

There are already some threads here showing macros that can do this for you. The search button will help you here. It has been a while since anyone posted about it so be prepared for older threads.

BTW: There is no specific macro that will work in all situations, so look around for one that suits your needs.