OP I don't think there is an easy answer to this. But I am going to tell you the story of My DS2.
DS2 was in Reception. He was the youngest in the year with an end of Aug birthday and emotionally mature for his age to boot. It's a small class with around 16 kids.
He had something of a love/hate relationship going on with another boy, we'll call him John. One minute they are best friends, the next they are falling out. The kind of thing you might expect from 4 year olds.
At the start of the spring term I get pulled in by the teacher. DS2 has bitten John. I am horrified; he has never bitten anyone before. I can't understand it. He's made to lose his breaktimes at school and at home he writes a letter of apology (actually he traces over my writing, it takes him ages). We spend a long time talking about how that is unacceptable behaviour. I know the mum to say hello to and I apologise too.
All is well for a few weeks and then DS2 bites him again.
Once again we talk to him, he writes a letter of apology, he loses privileges.
Everything calms down, DS2 and John are getting on well.
About 6 weeks later I get the dreaded 'could I have a quick word?' from the teacher at pickup. John has given out invites to his birthday party. Everyone in the class has got one except DS. DS has been distraught all day. The teacher spoke to John's mum to ask if his invite had been overlooked, mum apparently glared at her and said 'my decision is final'.
I spoke to DS, I told him that John's mum didn't want John's birthday to be spoilt and she was worried that DS might bite John again. I told him it was a shame but (not in so many words) his own fault. I told him he must treat people nicely. DS understood.
It didn't stop him crying himself to sleep for weeks.
When it came to his birthday, he told me he didn't want to invite John because John hadn't invited him. I told him we aren't nasty, we don't want to exclude anyone because look how upset he had been when he was left out. He agreed and we invited the whole class.
I didn't expect John's mum to accept, but she did and John came with his Dad. We had the party at home. It was a great day and everyone enjoyed themselves.
Since then DS2 and John are pretty much the best of friends, occasionally falling out but mostly friendly. They have had sleepovers and been at each others parties. In fact DS2 is one of the 2 children invited to John's 9th birthday treat.
DS2 has never bitten anyone else. But he learnt a very harsh lesson early on.
John's mum, on the other hand, has never really lived down the fact that she was the mother prepared to leave one child out.
I think on balance your decision has less to do with Frank and more to do with his parents. I would try talking to the school about the behaviour and see if they can have a word with the parents. It might be easier to suggest to the parents that they stay and supervise Frank if the school have made them aware there is a problem. But whatever you do, please don't leave poor Frank as the only one not invited. I've seen what it does to a little boy and it's heartbreaking.
Crikey that was long, sorry 