Makes no sense. If you have simply switched policies when one expired, then how can you backdate anything? In fact its illegal to have 2 policies on the same car. So if you can prove that there was an existing policy on the car from May to Sept then Kwik fit have no case. Even if you did not have previous insurance, its your choice as to when a new policy starts. Ask to speak to a manager rather than the telephone answering monkey.
I had fun with them this year

The tinkers liked to call in direct debits a week early when no money was in the account

Also at the end of the policy year, I canceled it (giving then 30 days notice in writing) and canceled the direct debit.
They then continued to carry on the insurance even though there was no direct debit setup

After 2 months of trying to take money off me and charging me for "so called" late payments, they sent the boys round to my house who was shown a copy of the bankruptcy ticket and other documents like my new insurence documents

Apparently, its a favourette game of theres to try and screw you for a few quid

If you can prove you was insured, I'd be tempted to tell them to stick it

Only problem is .............. by now, Kwick Fit would have registered your name with Equfax as a bad dept
