What type of underlayment should I use for an above ground pool

The first issue you need to address is why is the base eroding in the same place time after time? The large majority of above ground pools survive many years with a leveled sand base. The probable reasons for your erosion problem is an unstable base, misguided water runoff or the base is a bit too high in relationship to the surrounding area allowing runoff to seek a path under your pool.

An XPS base may help a lot, but even with a foam board base it needs to be on a stable compacted base. If the foam is installed over an area that is going to erode too deeply, it will collapse into the void created by that erosion. I strongly suggest you analyse the drainage around your pool area and correct or redirect runoff before you destroy another liner.

What you see there is not just water - it's probably already an ecosystem all by itself. If you would examine it, you'll see that there may be thousands of bacterial and other microscopic species there, most of which you wouldn't appreciate living in close proximity with people.

To make sure your new renters aren't killed off by the pool creatures, please make sure it is sealed closed, with no risk of people entering it, falling into it, or otherwise being able to use it. This, of course, assumes that you need it to function as a pool once your renters leave. If you don't care for the pool at all, simply fill it in completely and erase all traces, and you should be good.

Or, you can always make it a fresh, pristine and proper pool, and charge your renters more for the privilege.

But as it stands, it's almost certainly unhealthy, and would constitute a public hazard somewhere.