Your JAA ATPL is now deemed to be an EASA licence. In order to have it re-issued as an EASA licence you need to complete SRG 1104 and as you rightly say, you need to get the medical and ratings page certified by the Head of Training of an ATO/FTO.
Yes. they should have that information but I suspect the process of looking up scans of all your SRG1119 rating renewals is a slow process. Quite why they need to see the medical when they now issue it from their own computer is a mystery. I am told you should allow 3 months for this process!