Hey Imperial,
I have transferred my license around Europe 2 times already. Each time, the SOLI transfer included both the license and the medical file. In one of those cases, I was the one who had to gather the last three medical reports from the AME's that signed my Class I renewals, and pass them on to the exiting CAA so they could pass it along with the license to the entering CAA. This was due to strict protection data laws in the exiting CAA.
In all cases, getting the new license from the new CAA meant the medical file was in their hands (no medical file, no license issued). So, if you have a Spanish license (either CPL or ATPL), you can carry on as usual with the Class I renewals. Whichever AME that signs will send a copy of the new Class I to your CAA.
Unless the French CAA has its own set of rules (knowing the French wouldn't surprise me...) I don't see why they are telling you that. Calling them and asking for clarification is possible?
As an advice, always ask for a copy of the medical report to your AME after each renewal, you never know when you are going to need it.