Interesting. I don;t usually have the Bluetooth icon on my Panel because I rarely use it on Linux computers.
So I added it, and went to the' Preferences > Theme and Appearance' Menu and changed the Icon theme form PixFlat to a few of the others.
it is not only the Bluetooth Icon that disappears. On a number of the Icon themes, several of the Icons go blank or got missing completely. Breeze Dark is one example.
So the problem is definitely there. but might not be an internal Raspbian problem. It is likely that Icon themes on the Repos might not have full icon sets correctly associated with applications installed in Raspbian.
Bluetooth for example, doesn;t disappear off my Panel. The Icon simply goes missing. If I click on the space where it shoudl be, I can turn Bluetooth on/off and pair etc.
Starting at the top of the list of Icon themes in Preferences with Adwaita, the only Icon themes that give me a Bluetooth Icon are:
So far, without rebooting, it seems that changing the Widget Theme initially got rid of my Bluetooth icon form the panel, but swapping the Icon theme back to one of those three above then doing the reload panel thing, gave it back to me, while keeping other elephants of the new theme.
I tested this on various Widget themes and reloading the panel after setting a new theme did NOT recover the Bluetooth icon, but reloading the panel after changing the icon theme (2 tabs right of 'Widget' tab) to Qxygen, Pix or PixFlat = DOES give me back the Bluetooth Icon.
I'm about to reboot now and see if I still have a Bluetooth icon in a theme called MurrinaAzul...
E D I T
Well, that was fun.
I swapped several themes and rebooted between each. As long as I stayed with Oxygen, Pix and PixFlat for my Icon Theme, i could choose whatever Widget Theme I wanted, swap to one of those 3 Icon themes, reload the panel and I had my Bluetooth icon. And even after a reboot, I still had my Bluetooth icon.
It might be worth checking to confirm on your Pi, but with any luck you should get the same result.
As long as you remember to make sure the Icon theme is one of those three, and then do the reload panel thingy - you might have the problem solved and can use whatever Widget theme you like.
Hope this helps. It was a fun little exercise