I play bass with a German bow, and teach this to my students. Recently my school aquired a 1/8-size bass. Should I get a German bow for this one, or would it be too big? Should I rather use a French 1/8 size bow, but with a "German grip"?
I wouldn't think it would matter. Both are valid bows with their own sets of valid techniques, and a larger object is easier for smaller hands than a smaller object. That's why we give fat crayons and pencils to young children. Smaller instruments have inherent issues with tone because they are smaller. That's true of any instrument, bass or otherwise. So, no, it probably won't sound the same to use a German bow on a ⅛ bass as it would on a ¾ or 4/4 bass, but that would also be true of the French bow. I would be more concerned about switching bow techniques partway through a child's learning experience and would therefore absolutely suggest sticking with the German bow if that's what you normally use. I rely very heavily on muscle memory with my students, and I think changing anything as basic as bow hold would contradict that. Take all of what I say with a grain of salt, though. I have tutored bass, but I am not physically capable of playing one because my left hand is simply too small and weak for me to hold the strings down correctly. I have enough trouble with cello, for the same reason. My primary instrument is violin.
You say you are a cellist primarily? For a long time, even though it is a completely different instrument, bass was primarily used to echo the cello an octave lower. The idea that the bass even needs specialized technique is relatively new when you put it in terms of the whole of music history. Therefore, there is a lot we are still figuring out and a lot that we are drawing from what has worked for other instruments, particularly cello, for longer than bassists have been doing it. Bass players have actually sought out cello teachers in order to better develop their bass technique. The Strad periodically runs articles on this topic, and if you want to do more of your own research you can probably find some of those articles online.
It's all about what your student likes. The seller might have their own opinions, but the one opinion that really matters is your student's. I tend to have a lesson or two where we try out the two different bows and go from there. Personally, I had an easier time with the German bow as I was starting out. Thought it was much easier to pull a nice big sound out of the instrument, and down the road I picked up a nice French bow and used that for some pieces as well. If I were in a position where I had to start off by teaching one style before the student had a chance to try both, I'd stick with German.
I would recommend against the French bow with a German grip, though. You can sort-of teach the proper German bow technique with it, but it's just not the same.