A.K.
I'm sorry to hear that you and your family are going through this, what a pain. My daughter had a blocked tear duct when she was a couple of weeks old but fortunatly it cleared up with some drops.
I wanted to let you know that I had a horse that had a blocked tearduct that had to be probed and flushed. I assume that the procedure is pretty much the same for humans as it is for horses. It didn't take very long and the horse didn't need sedation. She tolerated it well and her eye was 100% the next day.
I know it's not much but I hope it helps to put your mind at ease. Best of luck for you and your family.