I would say you have to much pinch pressure. You say you never had this problem when playing guitar, then you really should not have it now.
Excess pinch pressure between the thumb and the fingers show up where you are feeling it Pinch pressure is different from grip pressure in that grip is a response from the forearms, where as pinch is a response from the thumb, the base of the thumb in particular where there are five groups of muscles to control the thumb.
Yes the thumb movement is supported in the forearm, but this is a laymens terms explanation, it is the control of pinch pressure that causes problems not the action source.
Anyone can make a tight fist or grip, you will feel it in your forearms, because of the ulnar side involvement, but make a tight pinch....now you feel it at the source of pressure the finger involved and the thumb with the thumb pad taking the stain, not the forearm as a whole just the radial side.
So relax that pressure on the pick, find one with a shape that suits you, maybe a bigger one to cover more surface area and make it hold rather than a pinch action.
Sometimes a lighter pick means you try harder to control it and get the sound, so try a heaver one to see if that makes you lighten up the touch, the heaver one requires less effort. I also say look at the Carole Kaye method, way way ahead of its time when you consider when it was written.
If all else fails see a doctor it maybe something un-related to bass but the task just shows it up.
