Yes, it would work and I've seen it happen plenty of times in cases similar to yours. I think some advisors were doing it in the past so they could get a brand new sale or brand new account in the bag, as this gets credited to their bonus.
Sky ideally want you to have one account for everything (it makes sense administratively, but also makes it easier/harder for the customer to cancel if products are not split between accounts), but if you don't mind having two separate accounts (one for LR and BB, and another one just for the TV) then you can do it. You'll be paying two bills obviously, but after the first TV bill you'll be able to align them for the same day (and you'll see two separate charges on your statement). Just make sure you remember you have two separate accounts in the future, as if you ever want to cancel everything you'll need to tell the advisor that it's two accounts (only one will appear on their screen and they'll need to search for the other account manually).
I normally does not matter if you use the same name, but in case the system sees an active account under your name when you're opening it online, I'd suggest you use your wife's name for the TV account, to be on the safe side.