I would find a RAD school and see where you end up. I would not do both schools as the teaching style may be different.
My son said the same to me about wanting to be a ballet dancer at 3/4. I ignored him and signed him up to football for a while as I thought he just wanted an activity to join in with (bad mummy!) Fast forward to now and he is on a pathway to audition for vocational ballet schools after we listened and signed him up to a very good local dance school when he was 7, which apparently is considered 'late'. He is passionate about ballet (we don't know why or where it all came from). So I would listen to your son and let him follow his interest. It'll fissile out on its own if it's not something he ends up interested in, but I'd give him the opportunity to have the best teaching so he has good foundations.
Regarding bullying, I would say my child did worry about it and asked to keep it quiet for a while, but in the end he has just decided he doesn't care what others think. As far as he's concerned, he wants to be a ballet dancer and if other people have a problem with that, that's their problem. He's only had one or two comments, which I think is mainly because he talks about it with confidence, which means other childern don't see it as a target conversion for unkindess. It's of course a risk, but so is anything (hairstyle, choice of clothes etc) and I would let your child be true to themselves.
I would also keep an eye on good associate programmes when he is older (Y3/4) if he is still interested then. Like royal ballet associates, tring associates, elmhurst young dancer associates, ballet boost...etc.
On a side note, you can afford both schools now....but if he pursues ballet, it becomes very expensive, once a week won't touch the sides. My son has lessons 4/5 times a week making up about 7/7.5 hours and that's considered nothing in comparison to some of his peers, who dance every single day, dance in festivals (we can't afford to do festivals anymore, nor do we have the time with our work schedules / commitments with our other childern) and have private lessons. And I do actually mean every day, even in the holidays.
One main school, with excellent tesching and good opportunities is the best strategy in my opinion.
Your boy only little, so enjoy the journey however long.