I had this nonsense with a VW Tiguan through Motability. VW set the cars up for standard or extended service intervals. Our car was set up for standard whereas Motability will only pay for extended servicing (less frequent). The dealer was able to reset the servicing light for me so that it wasn’t illuminating.
There’s no point in asking them to put anything in writing. It’s a lease car so whatever goes wrong is down to them anyway, regardless of the cause!