OK, it sounds like their policy is that if you've booked a seat for an infant and you want them to use that seat when the seatbelt sign is on, you'll need a car seat conforming to their guidance. If you don't have a car seat, or you bring a car seat but it's not appropriate to use on the plane, they'll put the car seat in the hold and you'll have the baby on your lap when the seatbelt sign is on.
That is a fairly standard policy. I wonder if it was explained badly to you as you should definitely be able to use the booked seat when the seatbelt sign is off, otherwise there isn't much point in being able to book it.
As for what car seats are approved, as you've said, you can't use your cloud z because it's rear facing which doesn't fit into their guidance, however, they don't require specific aircraft approval, just the standard R44 approval. This is helpful, because the only aircraft approved forward-facing seat is Britax Eclipse, and this will be no good to you because it only goes forward facing in a car as well. At 9 months I would definitely not recommend that and I think it's illegal anyway in most if not all US states. Plus, he might not even be 9kg yet at 9 months (some babies are, but only about half of boys and 1 in 4 girls).
Soo.... you could look at belt fitted rear facing/forward facing seats which might fit into their guidance. Just from other people's anecdotes, Joie Tilt, Joie Steadi and Britax Two Way Elite fit the bill there (although I haven't done it myself). The only way you might fall down here is that technically these seats need to be fitted using a 3 point seatbelt, not a lap belt, and the CAA says:
"Child restraint devices can only be fitted on a suitable aircraft seat using the type of seat belt/harness for which they were designed. For example a child restraint device that needs to be secured by a car type seat belt which includes a shoulder harness, cannot be installed on an aircraft seat that is fitted with a lap belt only."
I think when people take one and use it anyway, they are pretty much betting on the airline staff not being familiar enough with the seat to question it, which is fairly likely, but in general, EU based airlines have the kind of attitude of: why on earth would you want a car seat on an aircraft?? So I have also heard of people just point blank not being allowed to use the seat they have brought. This is different to North American based airlines, that have a specific blanket policy saying if the seat is FAA approved, nobody is allowed to say you can't use it. However that may also be because there is no baby seatbelt on those flights so babies with no car seat are completely unrestrained.
I think if it was me I would probably not bother with the car seat in the cabin, just use the normal baby seatbelt they give you for your lap, and maybe ask about booking a bassinet/skycot. If you need one for driving when you're there, especially if you're likely to visit the US again, look at FAA approved US car seats, order one to an amazon locker at the airport, you could probably use it on the way back if you do this. Or buy a cheap/lightweight one from the UK or bring your current one and take it in the hold.
Personally, I probably wouldn't recommend trying to travel with whatever will be your main car seat from now on, mainly because if you're looking to switch at 9 months ish you definitely want to stay rear facing at least initially, and the lightweight rear facing seats are a pain to use but the nicer to use ones are heavy and bulky. You can get some more compact forward facing seats that might travel well but most states require rear facing up to age 1 (some up to age 2) and again, wouldn't recommend that for main at-home seat from 9 months.