@eggman I think you can force a specific order for the sound cards through configuration, described in this Archlinux wiki page. You just need to know the name of the sound kernel modules loaded, but that's easy - you look in /proc/asound/modules:
pi@retropie:~ $ cat /proc/asound/modules
0 snd_bcm2835
This means the 1st soundcard (index=0) is controlled by the onboard PI sound chip, with the kernel module called snd_bcm2835. Your output will have 2 sound modules loaded, so you can note the name of the USB wireless sound kernel module and create a file ( /etc/modprobe.d/0-sound.conf) with 2 lines:
options snd_bcm2835 index=0
options <NAME_OF_THE_2ND_SOUND_MODULE> index=1
Ideally you can try to replicate the problem - sound routed to the wireless card - then see if applying this configuration fixes the problem.