After ironing out various minor failures, I was able to test an "operational" heatsink with my test rig. To my surprise I was able to push PLA through the heatsink, into the heater block, and out of the nozzle. However, it quickly became harder and harder to push the filament. On a few test pieces, pushing too hard cracked the heatsink at the point it connected to the heater block.

So why was this happening? Well it appears that the high-temperature resin is not particularly slippy and the semi-molten PLA is very happy to adhere to it. As it does this, the path of the molten filament closes until it become impossible to push filament anymore.

One traditional hotend uses a PTFE liner. So a solution to the sticking problem might be to coat the inside of the filament pathways with PTFE (which you can buy as a spray). Experimenting with this did improve the performance, but it only delayed the moment when the pathway became jammed. It's not clear if the PTFE coating didn't stick to the resin effectively or because of some other problem.