Vogue and The Cut independently place Lady Gaga at The Carlyle for T Magazine's October 19, 2016 celebration and document her performance. This verifies an event, not a stay.
Lady Gaga performed at The Carlyle in New York during T Magazine's celebration of its 2016 “The Greats” issue. Vogue and The Cut independently place Gaga at the exact hotel and describe her performance there on the night of October 19, 2016.
Source conclusion:
The cited sources confirm attendance at an event at the hotel, but not an overnight stay.