BE95 Flight Instructor reported shutting down the #1 engine at 13000 feet msl and attempted to restart it several times. At 10000 feet msl; they realized the right engine alternator was not working; advised ATC; and landed.

Narrative

BE95. Training flight. This includes shutting down one engine completely and restarting it. We obtained Class B clearance from ZZZ Approach to go up to 13;000 msl and descend down through the 12;000-10;000 msl layer if needed. We positioned ourselves about 3-4 nm SW and W of ZZZ1 for safety if we had any issues. At just below 13;000 MSL; we feathered and secured the left/critical left engine and then attempted to restart the it four or five times; losing about 3;500 feet. As we reached 10;000 msl; we realized that the right/operating engine's alternator was not working. We were worried that the plane's battery would not last long enough for us to get down and land with radios at ZZZ1; so we [requested priority handling] to get into ZZZ1 quickly as there were a lot of planes in the pattern and a couple of planes on approach.The descent; approach; and landing were normal. We taxied off the runway with one engine; parked and were able to restart the non-operating engine. We were not able to replicate the alternator issue.