How can I prevent an above-ground pool underlayment from failing due to erosion at one location?
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.