I took apart a relay to use the coil and see if I could focus the flux through a small screw head to clear the bits. It didn't work, although it did set bits, so I know it had a fairly powerful field strength.
At this point I'm going to abandon the quest for clearing bits with an active stylus. I have a plan to "solve this problem with firmware." It has the benefit of eliminating the need for a potentially cumbersome active stylus, and it will help add some "color" to this project. And I'll still be achieving the [virtual] end goal I had in mind via a different path than I originally set out on.