ESA medical certificate

I sent an application for ESA. For a few reasons, it has been backdated to the middle of last year. I've recently moved, and as a result, have changed doctors.

I got a medical certificate from the doctor from todays date covering the next 6 months, however my doctor would not backdate it as I have only recently registered with them.

I was born with a disability, but am under a specialist hospital so have never visited the doctor before today, even though I was registered with a different doctor/borough.

Will I no longer receive the backdated money as I cannot supply a doctors note for that time? Is it worth writing to them with proof of my disability etc, or is it pointless?

Thanks.