Wait like official surrogacy or like unoffical forms of surrogacy where sex happens between the one giving birth and the one contributing semen?
Depends on what the characters would prefer tbh. But given this is such an obsessive couple, I doubt either of them would really be happy about the actual intercourse-type. XD
So, artificial insemination it is, I'd say.
I don't think it's just the need to have offspring per se but the need to have children with their chosen "mate."
Having someone else carry the child to term and give birth might not have the same effect as being able to have a family with the person their whole existence has chosen to be with till death.
You have to keep in mind that they don't think or function the same way as regular humans do anymore. We might be ok with surrogacy/in vitro pregnancy, but I don't think animals would be able to get that same sort of satisfaction instinctually.
Just saying but...wouldn't it be a worth a shot to get a lady that offers her womb and see if that sates the instinct of procreation?
Just like gay couples ask a guy or girl to borrow their respective reproductive organs, these two could also give this a try.
I don't care much for seeing stories about couples with children (they just waste screentime of the couple 99% of the time and drag stories into slice of life genre, which I cannot stand, boring af) but once in a while it can work and IF this will make both of them happy: Get two ladies and make a deal with them to have offspring. See if you can all be happy afterwards.
irl I would loathe this idea, but fiction can totally go for this. I think it could be interesting. Especially if we involved a femaleXfemale couple for this. Would be 2 birds with 1 stone. XD