For anyone with the same question I had, I went ahead and bought a 5-micron filter bag and fastened it to the pump's water output yesterday afternoon with the pump cartridge removed. By the time I got up today, the water was clearer than it's ever been since we inherited the pool. I could even see the sand on the bottom, which I will be going after soon with the same filter bag and our cheap Intex "pool vacuum". Granted, the water wasn't completely green when I started, but this $6 (plus $7 shipping) bag has done more than a cartridge ever would have, and with virtually no maintenance.

Now, it feels like there is actually a point in maintaining the pool since I have found a way that is almost easy enough to do with your eyes closed and is easy enough for me to do all by myself.