How do they determine what surname the child of a contract marriage gets? Do they take the surname of the parent whose Clan is going to get the child?

That would make sense, right? This is all spelled out in the contract, well ahead of time. The participants of the marriage know going in which Clan will nurture the child. Anything else would be cruel.

So, since Tiazan, say, does not look to Korval, the child of a contract-marriage between a Tiazan and a yos'Phelium, in which the child is to came to Korval, would be named thusly: New Child yos'Phelium Clan Korval.