The Intex SWG isn’t a bad option for a small pool like yours. They have two, one rated for pools up to 7,000 gallons and one rated for pools up to 15,000 gallons. I would suggest going with the larger one because the general rule is to get a SWG rated for twice the capacity of your pool. The bigger one is around $200 on amazon, so well within your budget.
They do also have all in one pump/filter/SWG units, however, I would suggest staying away from those. The big disadvantage with those all in one units is that they don’t have separate timers for the pump and the SWG, so that can make it very difficult to balance the amount of chlorine you’re generating with the amount of filtering your pump is doing. By using a separate pump and SWG they will each have their own timers and the pump can be run for longer than the SWG.