The newer version of basswasapi.dll did not make it work. However, I did manage to get something in the error log after swapping it. I only see the one log line, even though I restarted MusicBee and tried to play files multiple times:
7/28/2021 9:53:58 PM - 10.0.19043.0 - 3.4.7805.33439D - System.NullReferenceException: Object reference not set to an instance of an object.
at #=zv0oS7QsmO66dgv$kZk96UXU4IA00.#=zsmYANNc=(#=zjsdVjkTpZ3Go$iKRt1PP6Wc= #=zQ3bgfJ0=)
at #=zGbZXiuqvCcFJ$rEc2sjoNJY=.#=zZEt6nM7AOo0I(#=zVzsNxnIpA1tczBaJkA== #=z3iMFQ3_g4XFi)
Swapping back to the original basswasapi.dll and trying the newer bass.dll did not make it work, either.
I am certain that nothing else is using exclusive mode at the same time.