I think ScotchOnDaRocks got it, but I'll say the same thing with more words. Since Troutulator doesn't do range vs range I just tested a set of one card draws against your hands. I checked both hands against a better tri and a worse tri with villain drawing to a Jack if you pat and both keeping any badugi if you break.
One nice thing about patting is your opponent has to draw to a better badugi. If you are both taking one he can pat on a King, but if you stand pat he has to go for at least a Jack, and more likely a T plus depending on his tri and strength at badugi. Killing two or three outs is nice.
If you knew with certainty your tri was better it would improve your equity slightly to break. Look how close a 267-tri is to a pat queen against a worse tri-hand. But if your tri is worse you throw away about 15% equity by breaking. Because of that I'd pat the Queen and the Jack here, only breaking for a nutted tri. (A25r has 5% more equity against a 6 tri than patting Queen badugi.)
Badugi sim, 100000 trials, 3 draws:
Dead cards: none
Hand 0: 2c 6d 7h Qs: standing pat
Hand 1: 3d 2s 8c --: 2nd draw: keep J badugi. 3rd draw: keep J badugi
Hand 0: 2
6
7
Q
: EV
58.8% - WIN/LOSE/TIE %: total 58.8/41.2/0.0 - badugi 58.8/41.2/0.0 - 3 cards 0.0/0.0/0.0 - 2 or 1 card 0.0/0.0/0.0
Hand 1: 3
2
8
-- : EV
41.2% - WIN/LOSE/TIE %: total 41.2/58.8/0.0 - badugi 41.2/2.7/0.0 - 3 cards 0.0/56.1/0.0 - 2 or 1 card 0.0/0.0/0.0
Badugi sim, 100000 trials, 3 draws:
Dead cards: none
Hand 0: 2c 6d 7h --: 2nd draw: keep K badugi. 3rd draw: keep K badugi
Hand 1: 3d 2s 8c --: 2nd draw: keep K badugi. 3rd draw: keep K badugi
Hand 0: 2
6
7
-- : EV
58.8% - WIN/LOSE/TIE %: total 58.8/41.1/0.1 - badugi 42.1/10.6/0.0 - 3 cards 16.6/30.5/0.1 - 2 or 1 card 0.0/0.0/0.0
Hand 1: 3
2
8
-- : EV
41.2% - WIN/LOSE/TIE %: total 41.1/58.8/0.1 - badugi 33.4/15.4/0.0 - 3 cards 7.7/43.4/0.1 - 2 or 1 card 0.0/0.0/0.0