One last update on this! I later found another forum where several people had had international bank drafts returned, so ended up flying to the US for a night and buying money orders.
For anyone thinking of doing the same, I recommend buying them from 7-11. I tried several USPS offices first, but none would sell them to me without a social security number, despite the USPS website saying you can buy them by showing a non-US passport instead. I also tried to buy via Moneygrams, but the closest eight(!) locations listed on their website as offering money orders actually did not once I got there. In addition, I discovered I couldn't buy money orders from Western Union because trying to purchase over $2000, even across multiple locations, triggers a system request for two types of ID, and as a UK citizen without a driving licence, I only had one. In the end, I tried 7-11 and they sold them to me in multiples of $500 without even asking for ID - all very straightforward, and they are Western Union money orders.
Something else to be wary of is that money orders can't be bought with credit cards, and for some reason, both of my debit cards kept being flagged as credit cards by the USPS payment system (before they realised I didn't have a social security number), blocking the transaction. The staff thought it was probably just because they were foreign, despite one card being my Wise international account. So, I ended up having to figure out a way to withdraw several thousand dollars in cash that day, which is a long story. It was lucky I did though, as the 7-11 I ended up buying them from only accepted cash purchases of money orders.
Basically, it was a lot of stress, but worth it to know I have the correct payment.