Did you state in your custody order that your child not be removed from the US without written approval or court approval?
You can't just remove his name off the B/C. Only in circumstances such as child abandonment, etc. which requires a certain amount of time, etc. before that can happen.
I believe a judge would take his threats very seriously, and require supervised visitation based on his threats. A restraining order is helpful.
Again, do not put his contact information on any school records.
You can try all you can to make it more difficult for him to do what he is seeking, but ultimately, you just need to keep CYA and be on the lookout at all times.
Get a free consultation from an attorney, if you think you may need more info. I did when my son was abducted 3 years ago and I believed my ex and his g/f were going to take my son out of the country. It was terrifying to say the least. I began a nonprofit org for missing kids after that happened, and have just taken one day at a time in case something like that were to happen again. At this point, my ex knows I would gain so much national and international attention to the matter that he knows he can't get away with it again.
Don't be scared, be smart and keep educating your self on how to stay safe.