That's frustrating about the GP surgery not doing the letter - I thought most surgeries would do it if asked, and many will charge for it (about £25), so it seems like kind of a missed opportunity for them to take a bit of money.
If you absolutely cannot get anything at all with his address on it before the 24th, then I would include the letter you do have along with a cover letter explaining why it is not dated in the last 3 months.
You could also see if you can get hold of anything else confirming his address after you have submitted the online application, since it will likely be a while before you actually need to upload the documents.