Sounds like things are coming along. Instead of backwashing, can you take out the cartridge filter and hose it off? There will be a lot of gunk to come out, and since you can't backwash, washing out the gunk will greatly speed up the process. I don't have one so I'm not sure how you you clean them without replacing them. I'm assuming you can wash it off and pop it back in instead of replacing it every time. After it's nearly clear and the bottom free of stuff, you replacing it would be a good idea.

Are you buying clorex brand bleach? If so make sure it is the plain ordinary bleach with no additives or "spring fresh", etc. You want straight sodium hypochlorate. If the is a Walmart near you, their house brand is usually the cheapest and doesn't have additives. (Just want to make sure you aren't accidently putting bad stuff in your pool!)

P.S. You WILL be able to see the bottom of the pool.