B.M.
If I followed your post correctly:
1. The father of your twins is Catholic and wants his kids to attend CCD and then the sacrament of First Communion in the Catholic Church.
2. The father of your twins is the man you describe as "Irish Italian"
3. You used to be but are are not currently a practicing Catholic and now are a member of the Orthodox church - where your twins attend services and communion with you on a regular basis.
4. Your twins have been baptized as Catholics AND ALSO as Orthodox?
5. In addition to the twins you have 3 older sons from your 1st marriage (or just 1 older son.... I can't tell, but you say 3 sons and then "older boys" so I'm not sure).
If your question is can your sons be taken by their father to attend CCD and First Communion in His church... the answer is yes.
If your question is can that same man take your OLDER son(s) - who are not his biological kid(s) - to the same religious education and ceremony..... the answer is no, unless he has a custodial arrangement.
It is NOT ridiculous for a parent to want to educate their child(ren) in the religion in which they believe. That happens to be protected by the Constitution. His religion is no more or less valid than yours and, as their FATHER, he has the right to educate them as he sees fit. The same as you are doing.
It's unfortunate that your kids are caught in the middle.
I wish you luck.