Personally I wouldn't like people around calling my son Bitcoin. It sounds like mockery. I believe when parents choose a name for their children, people have to respect it, and call them by the right name. However, I think the one who has to say if it's fine or not is your son. Does he feel uncomfortable being called Bitcoin, or does he like it and feel proud? If he enjoys his nickname, probably there won't be any issues at all, as he won't feel bad about it futurely. Maybe for you it has a special and positive meaning, since it was through BTC his childbirth was paid.
It might not be mockery but the circumstances behind the birth of the child. Sometimes you might not be able to control the nickname people call your children as much as it is not a bad name. I have heard several funny nicknames given to children and the parents will just laugh over it. The only problem the OP has is that people might see him as wealthy and this could lead to security concerns. In my area, people don't know that you can buy Bitcoin in bits so they will easily assume that you own one bitcoin which is a huge sum.
The mistake OP made was to expose how he paid the hospital bills for the child's delivery and they now had to spread the news. I wouldn't also blame him because he didn't know it would turn out this way. If I were OP and had the means, I would relocate from the location because it might be risky.
Now that everyone in that area already know OP is a Bitcoin enthusiast, I fear there isn't much he can do... He could say it was an isolated event, where he used all the BTCs he had to pay for the childbirth, discontinuing further purchases of Bitcoin later on, but since he is constantly proudly speaking of Bitcoin to his community, and teaching them the basics, how to invest for the long term and so on, it would be hard to make them believe he gave up on BTC investment...
If he has enough money to move to another area, it would be a good idea, like you suggested. Preferentially to a place where being a BTC enthusiast doesn't put a target on his head immediately, although it's easier said than done, because people have deep roots on the areas where they live and have grown side by side with their families for different generations. To leave everything behind is a hard decision only very detached individuals are able to do. Now he has to measure how risk it's for Bitcoin Junior to live and grow up in this place, having everyone around knowing they deal with BTC, especially after next bull run, which is going to boost BTC's price to even higher price ranges.