I had almost the same accident years ago, except my horse fell through a jump (triple combination - cleared the first, misstrided, knocked down the second, panicked completely and fell through the third. He was not the sharpest knife in the drawer.) Broke my pinky in 3 places and tore all the tendons. Had almost the same cast you did (except the other way around). They had to surgically repair it, put 2 stainless steel screws in my hand.
The good news is that, aside from the fact that the finger is ugly (it has a permanent crook in it but the scar is no longer visible), it's 100 percent functional. I can sew, type, and play the piano.
If they want to surgically repair the tendon, I would say go for it. It really wasn't a big deal (and I say that as somebody who gets spectacularly sick from general anesthesia. It was still worth it.)
When I went to see the orthopedic guy yesterday I was really hoping that he would just decide to go ahead and do the surgery first, rather than have me wear this cast for 6 weeks and THEN find out that I need surgery and have to have a healing period after that, but such was not the case. He just put a better cast on it and said come back in a week. :-/
Did you have to wear a cast or splint after your surgery? How long did it take before you could really use it again?