So I have some work to do porting the OpenCL kernels from bullet3 to work on ATI cards in Ubuntu before more progress can be made implementing the plan @Steven Peters outlined above, at least on my particular setup. I'll report back when I get this working.
Bullet3 is now working fine on my old HD 6900 ATI, so I decided it was time to follow @Steven Peters suggestion and increment the minimum required bullet version to 2.83.
It appears to be working after I made a couple of minor edits. I can make a pull request if anyone else wants to see what I have changed so far. I'm pretty sure I haven't begun to scratch the surface of the changes that need to be made to fully utilize the GPU computing capability of Bullet3, but first things first, Gazebo works with bullet-2.8.3.