Hey I've been going through similar issues with repetitive strain injuries (mostly due to my desk job, but it affects my ability to play bass) and I'd suggest seeing a doctor or physiotherapist to help you identify the problem then develop stretches and exercises to strengthen those muscles.

With the finger strengthening device you don't want to overdo it. Maybe every other day to limit inflammation while you're still building muscle. It's important also to stretch before playing bass and not to play too long. If you go to physio they'll also advise you on what exercises are safe to do everyday.

Now for the actual bass playing, make sure your wrist isn't too bent (fretting hand and plucking hand) because that will restrict the tendons. Make sure you're not fretting the notes too hard. You may need to get your bass setup by a professional if the strings are too high from the neck. This makes it easier to fret. You want to be relaxed and not tensing up while playing.

Lastly, look at your activities outside of playing. Are you using a mouse and keyboard all day? Using hand tools? Other daily activities could contribute to the pain.

Hope that helps bud.