Trevor, I will share what little I know with you.
The "cliassic" test as I've heard from others is that you stand up with your hands at your sides - if your palms face inward, toward your leg, you bow German; if your palms face backwards, you bow French. NB: This probably belongs in the "old wives tale" category but hey, you want to have heard all the old wives tales, too, right?
The only way to know for sure which is going to work better for you in the long run is to spend enough time getting proficient at both of them - there just isn't any substitute for that.
The thrust of the other replies in this thread is spot on - we have a saying in the gym when people ask if squatting is bad for your knees. The answer is that squatting isn't bad for your knees, but how _you_ squat may be bad for your knees. The same holds true for how you bow, whether German or French. Good to look at how a lot of good players hold and move their bows, good to ask for advice here - you'll find a thread that I started that included pictures of my German bow hold and the replies I received caused me to completely rethink my grip, and it's for the better: I sound better and my arm is much less tired.
Don't rule out the need to stretch. Even though your grip ought to be completely relaxed, life isn't perfect, neither is my bow technique and likely neither is yours, so find what gets tight and see about stretching it a bit both before and after you play, especially after.
What I know of kinesiology suggests to me that, if anything, a French bow hold is more likely to cause problems than a German one, since more people have issues with too much internal shoulder rotation and external rotation generally doesn't bother anyone. The French hold is the more internally rotated hold for your right shoulder. That bit of knowledge and 50 cents will get you on the subway and nothing more, however - the amount of tension being applied should be low, and because many, many, many people can play for hours and hours and hours with either hold without discomfort or injury, you need to look into the specifics of what you're doing.
A fine place to start would be to take a short video clip and post it here. We're not doctors but we might see something that could be helpful to you.
Just my opinion, worth what you paid for it here on the Internet, and if the discomfort is anything more than very minor, you should make your next step a visit to a qualified medical professional - I'd start with your regular doctor.
-S-