As you have a UK passport, I believe your embassy no longer issues that letter with a new passport. You don't need that letter. It might be a good idea to bring the passport issuance receipt if you have one. There was a report a while back someone asked for it instead of the letter. Usually, the immigration office knows which nationality can have that letter or not. So it wouldn't be an issue if you don't have one.
I don't know where you saw "letter from embassy (if needed)”, but I could say "if needed" is not so correct explanation.
, tourism extension is only once per tourist entry (on visa-exempt or tourist visa). But medical reason is different and you can apply for it with proof.
You talk with your hospital and doctor to see whether they are able to prepare documents for you to apply for an extension for medical reasons. You need to apply for this extension BEFORE you go on overstay. Once you are on overstay even a day, you cannot apply for a medical extension.
Medical extension is 30-90 days depending on the doctor's directions.
Otherwise, you have no choice other than to overstay and pay the fine when you leave Thailand.
First, you need to check with your embassy in Thailand if they would issue an income affidavit. Some immigration offices don't accept the monthly transfer method when your embassy issues an affidavit.
When you are sure you can use monthly international transfers to meet the financial requirements, you need a bank letter that says you have your bank account and 12 months transactions history.
Remember, You still need to have a balance of 400K baht to meet the requirement for your current stamp.
This time you need to prove both of these.
- 40K baht remaining balance: to maintain your current extension valid
When and IF you are asked for proof of funds for tourism entry, it has to be 20K baht or equivalent in major currency in CASH. The bank book doesn't work in that case. Be prepared with proof of funds, accommodation, and onward travel.
You have been in Thailand on the right Non-Imm visa and extensions, not as a tourist, it might not be as so-called "extensive history travelling Thailand" that much.