Somebody has already phrased this better but...
CB should be paid to the person who is the main carer of the child. Although in some circs you can agree for this to be paid to someone else.
CTC/WTC should be paid to the main carer, but again you can agree for it to be paid elsewhere. It's not a requirement for TC's to be paid to the CB claimant, but the criteria for both are the same so in theory that's how it should be paid. (It was previously a requirement though).
I work in this kind of area...[vague emoticon]
The reason you can have it paid elsewhere is because no family is straightforward, but mainly to help foster carers, guardians etc.. ie people who help out when there is some sort of family breakdown.
BUT.. that leaves the system open to loopholes. This is one of them.
I'm fairly sure that the definition of being a couple for TC purposes is "if you live together as a couple". Not live under one roof as a couple. The OP is a couple. They live their lives as a couple. Therefore they should claim as a couple.
My dsis DH is in the army, based in barracks miles away. She is a trainee nurse so couldn't move with him. Sometimes they go stay with him, others he comes back to theirs. Sometimes they can go a month without seeing each other. They are still a couple for TC purposes.
There are two issue in this for me. While I'm here debating whether they should be classed as a couple, that's not what the OP is asking. She wants to know whether she can defraud the system a bit more by putting the claim in the name of the lowest earner in order to defraud it a bit more. So does it matter whether in the eyes of the TC office whether they are a couple or not? No. Does it matter morally? Yes. Whetever the specifics of their living arrangements they should be claiming as a couple. Any "couple" who can separately afford to pay two mortgages does not need any extra help from the government.