Thank you for all of the replies!

To touch on chemical stripping, the rest of the car is painted. I'd like to keep that paint, at least for now. Splash would definitely be a problem.

The car can be inside or out without much trouble, it's on casters and I have a flat driveway.

The engine bay isn't dirty, but with welding up a bunch of holes and grinding the welds, it seems as if I'd be better off stripping everything and laying down fresh primer.

Seems like wire-wheelin' is still my best bet. I think the one I was using was a little too soft bristled/worn out to be very effective. I'll grab a new one, or three and get after it this weekend.