Ubuntu recently came out with Ubuntu Studio. Ubuntu was one of those distros that could run on a Turion. If Ubuntu can work on this, Bodhi cannot be far behind. The Broadcom wireless can be overcome, as can be gleaned from our own forums. The Radeon was giving the black screen on boot issue; this was not going to be easy. But I checked to see what should work on this Turion chip: Ubuntu, SLAMD64 (A Slackware spinoff), Manjaro, etc.
But TAILS turned out ready to run with both the wifi and the Radeon graphics, out-of-the-box. If I weren't looking for other laptops--a friend needed to run special auto diagnostics with this gadget, I would just nuke the existing OS and install these distros. It's a play-it-by-ear thing.