It just seems pretty suss - doesn’t it seem that way to you? From the little you’ve said I would question wether he has really broken up with the ex, whether he is doing anything at weekends other than looking after the twins.and whether he does actually want to live with you full time in a joint-finances-together-forever way rather than just a cocklodger way… but I’m sure there’s lots of background info you haven’t included here. it is at least reassuring to read that you’ve met his family and friends etc, so maybe it’s not quite as peculiar as first it seemed… I presume you have also been to his place and spent time there, so you know for sure that he does have his own place…
Regardless of this, I can totally see why he would want to reassure and prioritize his 3 year olds right now, and be available to them when possible. From their point of view (if they were old enough to figure this out) their dad spends over half the week with his new baby/new girlfriend and two other children. They get 48 hours a week. No harm for him to make sure they are prioritised at the weekends while your baby is still so little.
Also - of course he gives their mum money every month. That is what he is supposed to do.
Plenty of people who do live together see each other barely any more than you currently see your partner, due to work commitments. Plenty people’s partners don’t pull their weight with a new baby, without having twin toddlers to also manage. If these were all your joint children you’d probably be delighted that he stepped up with the toddlers at the weekends. He is only one person and can’t be in two places at once.
If he doesn’t participate when he is with you then talk to him about it… Does the baby sleep through? Does he help with your older children when he is at yours? Or deal with the baby when he gets home so you can spend time with them? How old are you older children? How do they feel about his twins? How do the twins feel about them? Have you talked with your partner about how it might work financially if you did all live together?
How are your oldest children handling all of this? Have they also met his friends and family? It must be incredibly tough on them… I would be much more concerned about them than your four month old if I were you, and be trying to take my focus off my boyfriend (hard as it may be) and onto my children - the ones who are old enough to be aware of the situation.