Shouldn't getrandom() be plenty fast these days?
arc4random has been provided by glibc 2.36 (2022), and is available on all the above-mentioned systems as well. If you don't want to make a syscall per request (outside Linux), just use arc4random; it'll be the fastest method available. musl libc lacks arc4random, unfortunately, but you can always ship a small wrapper.
Systems that support arc4random also support arc4random_uniform, which is a way to get an unbiased unsigned integer between 0 and N (up to 2^32-1). That's probably the most important reason to use the arc4random family.
I think neither are unbelievably slow. I dunno, take some measurements and see, maybe it suits you.