Culture clash - you said it! This is an ALMIGHTY culture clash 
I didn't realise I even had a culture until I married into another culture. I take it your first child is a girl? Did they pierce your daughter's ears the second she came out of the womb? I had my PFB girl strapped to my side for a year so my ILs wouldn't whip her off and get her ears done. I gave in, eventually, when she was 4. I just couldn't take the bloody pressure - on and on and ON.
I don't know what you're going to do here. It's all very well posters saying 'no way would I do that!' but they clearly haven't married across cultures. It's very different when you're in it. This stuff runs deep , as you have seen. One culture isn't better than the other - but, sadly, shocking bullying does come up if a cultural practise is threatened.
Your husband obviously doesn't realise how deep his culture goes in him. he has behaved very badly - but he is being crushed by shame, obligation, guilt. he probably doesn[t realise what he's doing, he's like a headless chicken. THAT'S NOT TO EXCUSE HIM, just that he's crushed between two cultures, two sets of obligations.
You should come first, yes - but in his culture it doesn't go like that and he will be having a really hard time reconciling the two. Terrible time for this to come up - particularly as you have had such a rough time having your lovely two babies. I'm so sorry you're facing this now.
I would have said (as I said repeatedly to my ILs!) that it is not in my culture to do blah-de-blah. I was speaking to the (willfully) deaf the majority of the time, but I still said it. Over and over again. I was also living in my own culture at the time, so I at least had a bit of leverage. I'm not sure you're going to be able to pull off ignoring their deeply-held tradition here - probably because you live in their country? Your husband is getting first-hand the almighty fallout from this - if he were in blighty he wouldn't be getting it as full-on, would be largely removed from it iyswim; relatively free to follow what you have both agreed together.
In our culturally-blended family, we have interchangeable names across the board (except me!) and no-one particularly notices. yy many people have the same name - but, somehow, we all know to whom we are referring - I don't know how that works, but it does.
I think this will be less of an issue as time goes on. I can't help thinking that, with this HUGE fallout, you're going to have to go with the tradition of the country you are living in. You can maybe sort out the difficulties as time goes on...
In the meantime, try to enjoy your lovely, healthy twins. You'll have tale to tell in the future! I do so hope you can get past this together somehow. Try not to let this awful thing eclipse the wonderful blessing of your healthy babies.