I've used an old T-shirt that I covered over the skimmer basket, sort of like a skimmer sock, and then added the CYA to it. After overnight running the pump, most of the CYA is dissolved and within one day (24 hours) it is fully dissolved. That's for adding around 20 ppm CYA to a 16,000 gallon pool. Adding more CYA will likely take longer.
I suspect that the sock is too thick to have enough water flow through it. The use of a T-shirt, skimmer sock, or maybe even panty hose would probably make it go faster. Of course this is partly because some very small pieces will slip through, but it will give you a greater sense of accomplishment! Actually, I've measured the CYA after it looked like it was dissolved and it usually measures at least 3/4ths of the added CYA amount so most of it apparently does get fully dissolved.
The problem with having it get caught in the filter is that the flow rate there per surface area is VERY slow which is why it can take up to a week to dissolve. It may be dispersed, but the water flow rate is slow. In my pool at 48 GPM through my 340 square foot area cartridge filter this would be around 0.2 inches per minute water velocity through the filter. Yes, you read that right -- that's what an oversized cartridge filter will give you. In the skimmer basket given two floor drains and one skimmer, I figure around 10-15 GPM through the skimmer which with a 6" diameter at its base would be about 7-10 feet per minute water velocity, though of course when there's a lot of CYA there the flow is very restricted.