Technically it is illegal to use UK certified seats in the US and there is no exception for tourists. But the reasoning for this is because some countries (e.g. Russia, some Asian countries) have minimal car seat regulations and there is some dangerous unregulated crap that gets sold there, so if there was an exception for tourists people could get away with using those. Rather than keep up with all the regulations and make a judgement about which are reasonable they just ban all foreign regulations, the same as we do here.
OTOH, there is an argument that it's safer to use your own seats because you already know how they work. Whether or not dodgy rental seats are a problem in the US, I don't know - they tend to have expiry dates on them over there, so they shouldn't be too old. But the US methods of install are different to ours and the manuals can be confusing which isn't the best thing to be grappling with after a long flight and possible jet lag.
The two problems I understand which might come up are that bracing on the front seat is usually not allowed (but there are no seats on the market any more which brace, so you probably don't have to worry about this) and support legs are not permitted for use in all vehicles - that one might be an issue. US cars do have LATCH which is identical to Isofix.
US seatbelts are different - they have two modes, the normal mode where the seatbelt locks under sudden braking, and a car seat mode where it ratchets and gets tighter the more you pull it out. To switch between the modes usually you have to pull it all the way out and let it click. You don't want to use the car seat mode with an EU/UK seat, our seats aren't designed for this.
Unless your seat is a completely different kind that doesn't exist in the US (like an impact shield type) it is quite likely that a random police stop (which is also fairly unlikely in the first place) would pick up on the exact type of seat, even though they don't have chest clips. Like most places, the police are generally happy to see people using car seats properly, because the major problem is parents who don't restrain their children at all. I suppose if they really wanted to make an issue of it they could write a ticket for it, I just don't know how often it happens in reality.
I've heard this "insurance invalidated" before but I don't know if it's really true or if it's just something people say and then it gets repeated because they have heard someone else say it. I've not come across an actual incident of it happening, which doesn't mean it never could, it just seems a little suspect to me. Considering a lot of car seats are incorrectly used and the majority of deaths/serious injury happen to unrestrained children, I would have thought it would get more publicity.
If you can get onto the CSFTL group on FB (the UK/EU one would be fine) that is moderated by CPSTs, which are American car seat experts, and they could probably tell you more accurately.
If you do decide to buy seats over there, it might be a good idea to choose the model beforehand, have it delivered to an amazon locker at the airport, and spend some time watching installation videos so you know how it works. Most US seats can be used on planes, so there might be an argument for this approach in that you can bring the seat back and use it for your child on the plane, but it might be a bit overkill.
Usually as said the car seat bags are a good option, you can usually pack them out with clothes/towels/nappies to protect the seat a bit, another I've heard is using the original box if you still have it or any large cardboard box of the right size.