Question about Approved Training Provider (ATO) in a different country

That depends a lot on the approving authority. In the case of an LBA-approved flying school, this is not possible.It is not even possible, if your approved base is German airfield A, to do many training flights beginning at German airfield B. They want to see at least the vast majority of flights begin at the school’s appraoved home base.

If by piggybacking you mean operating a satellite training facility then yes if the competent authority permits it. The satellite need not be in the territory of an EASA Member State either. Naples Air Center in Florida operates as a satellite of Andrewsfield Aviation and is therefore not under direct EASA oversight.