Here comes a short update.

I have tried the squeezelite default settings again (plus -D for DSD) and now all sample rates as well as DSD64/128 are working without the need for server side upsampling.

Ubuntu 22.04 with current kernel 5.15.0-47-generic.
Don't know which was the version that fixed it, but the current one works.

So finally the whole issue was a Linux kernel bug ;-)