Any EASA ATO can issue you with a Course Completion Certificate.
Re the medical, any EASA medical certificate is valid however; the details would have to be sent to the medical dept of the State that issued your licence. As I understand it, the German authorities have a data protection issue re sending certificates to another State. Nothing to stop you photocopying it and sending it to the CAA.
I deliberately didn't answer the medical question as I knew there was an issue which manifests itself if you try to change your State of issue from Germany to another State.