You could get creative with a dremel tool; grind some flats on the outside of the round head. You could cut the head off if you have the bolt loosened out enough, but then the issue is getting the axle off the remaining bolt, probably would not come off without moving the diff over or moving the swing arm (big job). You could try to drill a hole sideways thru the bolt head and stick a hardened pin thru it and find a way to grab and turn it (but it's hardened steel). You could use lots of Liquid Wrench and heat - probably should do that no matter what you do. If you can devise a way to turn it, try to work it back in, then out a bit - back and forth until it loosens up a bit.
Huge vise grips and the above penetrating oil, heat, and loosen/tighten routine.

As to the problem, you might have a center driveshaft bearing and/or u-joint bad. They can definitely act differently whan turning one way of the other.
The bearing is easy, the u-joint is $350 minimum. When my u-joint got old at 200k it vibrated slightly only when accelerating thru about 15-18 MPH, so maybe that's not it.

DG

DG