I ended up answering to my sisters name at school, as well as my own. For some reason teachers couldn't tell the difference (even though we had different colour skin for goodness sake! Not to mention the dramatically different length hair and two years apart! I didn't mind that much, just rolled eyes and laughed about it at home.
You get used to answering to a whole variety of names as long as you hear them often enough!
The ultimate irony is that she changed her name when she went to uni (a very different sounding derivative of her name, which doesn't have even 1 letter the same as her usual nicknames growing up!). So I was being called 'her' name at school when she wasn't being called that name herself! Ah legacies... :)
My son has a first name which he definitely recognizes as his own, though we don't use it day to day. We use his second name which is the western version of his non-western first name. It ended up that way due to stbxh family getting very upset and demanding about names when DS was born.
We settled on this way round it all, and still have lovely names for DS and he will have scope to decide what he wants to be called as he grows up. His first name crops up on registers and work books from time to time, but as people can't pronounce it properly anyway, he doesn't get shouted at for not answering to whatever random sounds people put together!
I think your answer lies in how you use the names... is Bear a family term of endearment? A baby name? An informal nick name? Or a proper name in its own right?
If it's more of a term of endearment than a proper name, then perhaps keep it as such. If it's become a proper name then maybe you should continue this usage in school too?
Will your DS answer to Jacob? Or will it feel a bit alien to him? If so I'd suggest a name that he really feels is 'him'.
Good luck deciding! And remember that although people have very strong opinions about names, it is your decision and for you to do your best for your son.