For your consideration: TSP is pretty nasty stuff. Make sure you're using correct PPE and try your best not to get any overspray on landscaping / vegetation. Also make sure you've rinsed them completely and the runoff doesn't end up in streams / lakes, or putting residual phosphates back into your pool. Algae eat phosphates.
Cleaning filters is something I actively hate, almost to the point of convincing the wife that we just need to chuck the cartridges and spend $200 every few months for new ones. But the cheapskate in me actively hates that idea. So I'm looking at other options. We're on well water and septic, with a stream in our back yard that feeds a community pond, so I'm not going to use TSP at my house. I've used blue Dawn dish soap in the same concentration (1 cup per 5 gallons) and soaked them overnight in a large garbage tub. Then I've tried several methods for rinsing them clean, and all of them aren't something I'm going to be able to do long-term without some runoff / erosion mitigation. Your cartridge system is likely way easier and more manageable, given the size of your pool.
The dish soap was very foamy and took a long time and a lot of messy water overspray to rinse off, so I'm considering trying using phosphate free dishwasher detergent (cascade, etc) or possibly even research if tide pods would work. We're already running those thru our system regularly anyway. I'd imagine someone on municipal water / sewer could easily and safely do the same. Most folks recommend either the single brass nozzle type sprayer or the aqua comb for the garden hose during the rinse cycle. I've had differing results with both. Seems like things go a little faster with the aqua comb. I used my small electric pressure washer on the first set, and I'm pretty sure that was a bad idea.