Yes, the more I look, I think it's just a strainer (to protect the
impeller from leaves). Sorry about "ionizer" - I meant ozonator. I've
been looking for filters on eBay, and all I see are lots of replacement
cartridges (for store-bought Jacuzzi's) or very expensive sand / earth
filters (around $600).

I want the plastic housing that the cartridge filters go in. There have
to be cheap ones out there, because I've seen inflatable pools that
include the pool, pump, and filter, all for around $50. Can I just hook
a cartridge filter up to a pump, and submerse the filter in the hot tub
(no enclosure)? What is really in the "guts" of the filter housing
anyway?