Has anyone had experience with dealing with a blocked tear duct in LO's eye past 12-months?

Our daughter (13-months) has dealt with this in her left eye since birth and it would often come and go. Our pediatrician wasn't concerned about it until her 12-month well visit, as she said most clear up by then. LO has never had an infection or any issues from this, but the pediatrcian mentioned she may refer us to an eye specialist after checking it again at her 15-month appt. I haved massage it gently with a warm cloth (as recommended, and now when she will let me) and it used to help, but not recently.

After doing some research, I found that a lot of times the first step is to probe the tear duct. I can't imagine my LO not flipping out if this were to happen. Does anyone have experience with this situation? Suggestions/tips?