I don't think that you have told us when you bought the car. However, you have referred to a conversation in which they apparently told you that the MOT had been carried out on 11 November so that suggests to me that you bought it after that date.
Although it seems as if you are dealing with quite a dodgy crowd, you may as well go through the paces of asserting your proper rights. Because you have discovered this issue within the first 30 days – you can add to the strength of your position by sending them a letter asserting a right to reject the vehicle under the consumer rights act.
If a car manifests a defect within the first 30 days then you are entitled to reject it out of hand with no chance of repair but you must assert your right in writing.
Send them a letter immediately – recorded delivery – informing them that you are rejecting the vehicle and telling them on what grounds and say that you are asserting your rights under the consumer rights act.
It won't make a whole lot of difference, but later on if you find yourself having to take court action, then it will all help.
Please let us know when you have had the AA check.
Meanwhile, I suggest that you contact me at our admin email address and let me know the identity of the garage and any other identity clues that you have unearthed. It may enable us to give you additional help
Assuming you're correct about the limitation running from the last date of deferral. The last deferral was in 2013 so the statute barring period would end on 31 August 2019, the money claim was made on 3rd June 2019 so is within the limitation period. Therefore the debt is not statute barred.