This subject can generate emotive replies. For the OP, please have in mind that the type of Renter's Insurance we are talking about is designed for US residents who have a US address and credit card registered to that address. It's possible to persuade at least one of the companies on the phone to accept a foreign credit card, but they will still insist on a US mailing address. It may be a good idea to make them note your foreign residency, but if you do, the premium may vary or not be available.
Some UK pilots use a convenience address like a hotel or the FBO. One broker told me that a hotel was OK, but the FBO not.
There doesn't seem to be any actual prohibition about insuring foreigners, although the brokers policies do vary on this. And there's no evidence of any payment being withheld on the basis of a claimant giving a convenience address. But in UK, you'd be very wary of giving an incorrect address to an insurance company!