Since water is present (the curing process uses water vapor), why not use use blue RTV gasket maker?
https://www.amazon.com/Permatex-80022-Sensor- … B0002UEOKK?th=1
(Passive google-fu says that acrylic plastic is not a recommended material for glacial acetic acid (the acid released during curing of this kind of silicone), but that it can tolerate 5% solution of the stuff indefinately. Many stories of people successfully using the stuff to make gaskets for acrylic surfaces. Repeated installations of it will likely cause fogging or crazing. If you can get proper gasket material and it actually seals, that would be preferable-- however, since your substrate is cracked or damaged, you likely need the extra bit of seal you can get with a silicone paste gasket.)
Basically, clean the water out of the gasket well, but leave the halves of the cooler wet. (The surfaces the gasket maker will contact MUST BE DRY. But the rest of the cooler being damp will improve curing of the silicone)
Pop the seal on the gasket maker, put on the stem, cut it at a point where the extruded silicone will be about the same diameter as your original gasket, but just a teensy bit on the heavy side.
Squirt the gasket maker into the groove in a smooth, even bead, overlapping the ends slightly.
Assemble. Put a weight on top to assure good contact.
wait 72 hours.
Use like normal.