Our first "Real" pool was an 8' SnapSet with sort of rigid plastic sides, just around 500 gallons, if I recall. We bought a small sand filter for about $60 at Wally-World and poked a hole through the side of the pool for the suction side. The return hose just hung over the top of the pool.
This was about 6-7 years ago, before the Intex or donut type pools had really hit it big. I don't think you can get the sand filter anymore, but you probably get one of the intex filters and do the same thing.

You do need to get some stabilizer into your water to slow the breakdown of your chlorine. You can use the tri-chlor in the floater until your stabilizer (cya) level gets in range (usually 30-50) and then just use the BBB method.

For what is left of this season, I would probably just dose with a bit of bleach for a few days, and then dump the pool weekly or so. Then I would regroup over the winter and decide how much you want to invest in an inexpensive pool.