Hi,
You had a contract with the garage. The contract was verbal and expected you to pay on demand between £150 -£180. You have show a gesture of good will where you offered payment upon retrieving your car and then again a week later. Due to their negligence your invoice was misplaced and you had no way of settling this payment. Debt recovery found your address ok so I am sure your garage had your address on file. In any court you are a winner.
1 - You had a verbal contract
2 - You have repeatedly offered to pay due sum.
Try not to get scared by debt bulling companies cos they want their share.
If the case does reach small claims court the garage will be liable for legal costs as based on your questions it was their negligence. You have just asked for what is legally yours, your invoice that you can demand upon buying goods or services.
It might be the case where the garage wanted to do cash in hand job without declaring to HMRC that makes this mater more interesting for Revenue Services.