I recently got my UK "automatic" license and did just rent a car in the USA, through Europcar. I used my UK credit card to secure the reservation, but paid for it at time of pickup in the USA on a USA credit card that has a UK mailing address. Renting as a European got me a Toyota Camry from Advantage at LAX for 9 days at a total cost of $212, including all the insurances with a zero deductible. I couldn't find anything remotely close to that by trying to book through Hertz, Enterprise, etc., directly. So I would advise going that route, if you can.
The continued validity of your USA license will depend on your state of residence, of course. Once I left California, my CA license was no longer valid. I could have driven on it in the UK for up to a year, but it would have been invalid in the USA. (This is not the same for members of the armed forces, who have different rules.) I don't know how the rental place would know that my CA license was no longer valid, as it still has several years on it until the expiration date. However, if I'd been in an accident and insurance got involved, things could potentially become really messy and expensive. With possible criminal charges for driving without a license, if I had not gotten a UK license and used my California one instead. As that potential nightmare would have bankrupted me, at a minimum, there was no way I was going that route. Hence the UK license - which I only actually got so that when I visit the USA I can rent a car!
So, suggest you check with your home state's department of motor vehicles licensing agency.