It was shared with me on Discord by quebecd3ltatw0 that there is now a solution for Proton as well!
Include WINE_SIMULATE_WRITECOPY=1 in your launch options.
I tested on Proton Experimental as well as UMU Proton 9 Beta 16 and it worked!
It did not work on system Wine 9.11, but that's fine. WINE_LG from PortProton continues to work if you don't want to use Proton for any reason.
Also I tested Proton Experimental in VR and it worked there too, which means as far as I'm concerned this one is "solved via workaround".
This is the same issue as
After the patch the plane loads on Linux now but HBUI does not work.
My guess is that whatever UI framework HB is using doesn't play nice with Wine/Proton, or we're missing some dependency or config option that is required.
vsTerminus's post in F-4 does not work on Linux was marked as the answer