Things seem to more or less be working now:
The issue was that I was forgetting to strobe the HMOVE register, which tells the 2600 to respect the fine adjustment value I was apparently correctly calculating and applying.
There seems to be a few little things going on with the HBlank, though. Maybe I need to split up the code calculating the delay and the fine adjust onto two different scanlines? It seems like some code is leaking into a scanline up there. My intuitions are still underdeveloped here.