I've had this issue a couple times. One time it fixed a gurgling pipe. First, be sure there isn't a piece of tobacco or built-up gunk or cake in the mortise. If not, you can very little by very little bit, use a fine file or very fine sandpaper perpendicularly on the tenon tip. Give it a couple swipes and then fit it again. It'll take less than you think, thus faster than you think. You can't put back what you remove, so it's better to go slowly and checking and checking and checking.
For me, it would depend upon the value of the pipe in terms of either money or sentiment. I ruined some really fine pipes by trying to do things myself, even though it seemed easy at the time. That being said, I would most likely send it to a pipe repair person or outfit of some kind and have them do the work if it all possible. With the proper tools it’s a three minute job. Trying to do things by hand, you could destroy a really fine pipe.