Are you 100% sure it is bursitis?

What you describe sounds a bit like it could be a start of thoracic outlet syndrome (TOS) or a related condition....MDs often don't learn much about TOS so ...

Has your cervical spine been checked lately, those arm/hand sx could be spinal related and that would need to be ruled in/out before looking into more into TOS.

Can you get any PT or massage therapy, possibly expert chiropractic?
Could even be trigger points/spasms causing the problem..I would try this first unless you think you might have some c spine issues, but often at the 1st appt/evaluation - PT /DC will test to check for c spine or soft tissue causes..

We have a thoracic outlet syndrome forum if you want to read more about it & various therapies.