If someone feels that strongly about the patch, then yes, they should go elsewhere.Talk to your teacher. Tell him why you don't think you should have to wear this patch. I'm a Christian, Roman Catholic to be exact and I would have a fit if someone made me wear a patch of any type that relates to religion (that includes Catholicism). I'm just not one for forcing your religious beliefs down someone's throat. If I had to wear the patch I would leave and go elsewhere.
I see the patch as being relatvely unimportant. It is the owners choice to make his marque the way he wishes. The gi is only worn in the school anyway.
This is not forcing religion down anyone's throat; the other student's pressuring and proselytizing is.
Also, if the students are paying (I assume that they are), then severing ties with the the organization where they hold rank (Bujinkan in this case) for personal reasons is very inappropriate and could be considered a breach of contract and bait & switch. Once he's broken with the parent org, the students future advancement is not recognized outside of the individual school. Part of why I train at a Kukkiwon dojang is because I wanted a rank that meant something more than just the school's say so.
Breaking with the Bujinkan should have been discussed with the students.
Daniel