Hi Casey,
taking lessons on an ATO aircraft of a 'not-yours' ATO has two barriers. First, the operational handbook the ATO has been approved on. I know of none covering this situation. So, you are outside SOP and they simply don't know how to handle that. Second, insurance of the ATO does have a word to this and they either don't know how to cope with it.
The only way I could think of would be to enroll at each ATO you plan to fly with. I don't know whether this is even allowed on current EASA regulations to be enrolled in more than one ATO? You may have to de-enroll each time from your current ATO, transfer all papers to the respective superior authorities of the next ATO - and this for every time changing place. The ATOs would have to synchronize their SOP and training syllabus, synchronize on medical and other issues, track your training progress in a way to keep their record audit-able andsoon ...
As was mentioned before - get time off, cancel appointments and do it at your home ATO is much more efficient and you save the pain of talking to different official concrete heads.