Turns out, there IS a solution, in the package #blueman.
It integrated into the #systray, but it kinds of stands out. I kinda like it - YMMV.
There. That was the only thing I could pick nits about! Awesome software!
PS - if you got here via the #systray hashtag, give me a boost so I know there are ~~ others ~~ out there.
#exwm now puts a nice background on the #systray: all the icons are
visible even with dark themes..
(https://git.sr.ht/~abarocio80/simple-emacs/commit/8f952b64102741d356b7dbf139ceef70e0270f91)