If you really look around (and get lucky), you might find a rosette cutter that has the kind of radiused edges that you are looking for. Of course, it will have some fancy co-centric patterns (that you won't want) inside that outer edge. If so, here's how I would proceed:
Clamp the bass body to the drill press table in the proper position for the hole you want. Use the rosette cutter to cut the edges, then, without moving the bass body, swap out the rosette cutter for a smaller forstner bit, and use it to flatten the bottom of your recess (or nearly flatten it; watch out for the lip that cuts a bit deeper with many forstner bits). Or, you could flatten the bottom with a router bit....
Another method would be to plunge cut with a dish-cutter router bit, which would leave a hole with a nicely radiused bottom but a right-angled top (right angle between the walls of hole and the top of the bass). If you wanted a radius up there, you might be able to use a brass-piloted 1/8" roundover router bit in a Dremel (mounted in a router base) or a 1/4" brass-piloted roundover router bit in a trim router (laminate trimmer) to round over the top of the hole. In either case, though, you will have to look around to find a bit with a short enough brass pilot (or you could cut or grind off part of the pilot if you have access to machine shop tools), especially because you want a shallow recess (because of your top).
A third method (which might be the most elegant) would be to clamp the bass body in position (very tightly) on the DP table, then use a regular old drill bit to make the through hole for the pot shaft. Then, without moving the bass, chuck an ogee router bit with the bearing removed into your DP, and with the drill press on its highest speed, very slowly feed the router bit into the body. The little stub-shaft that the bearing usually lives on will feed harmlessly into your pot shaft hole.
All of these techniques can yield different shapes for your hole recesses; you can play around and find something creative (and not exactly like PRS) if you are so inclined.
