I would have, if I was GM'ing, just had the girl tag along.
I mean, how hard would it have been to give in on that little point and, as you mentioned, she is traveling with gunfighters. How many hit-points can she have? A stray bullet in the first encounter to come along solves the tag-along problem and gives the moody character something else to be moody about and move the game along. Or to be nice, you find someone willing to take her in at the next town. Just because you aren't prepared, don't punish your players. If you really aren't prepared to deal with it, just say so. It's better to move on as has been indicated before in these forums.
In general, when dealing with stubbornness, just try and make it work as well as you can without killing the game. Don't stop to deal with it, just handle it on the fly. It may present itself in a good way later on. If you stop, you are giving them exactly what they want - attention. And not in a good way.
I have had an instance where a player really wanted to do something. It was for no other good reason than he was trying to be a shit. So I let him. No one else gave a crap, that player had their little fit and the rest of the adults at the table let him know he was being childish and I punished his character in the next encounter. I did it in a way that was a little harsh, but still fit the game - not like a lightning bolt out of the sky, but if you have most of the baddies focus on them because they made themselves the target for a good reason, it can still work. They will either learn, or keep getting burned. If they get upset, maybe they need a hobby that allows them to re-load their last save and try again for a more favourable result...
Gm'ing is on-the-spot. If you plan for everything, you will never play. I would rather game with less prep than not game because I wasn't totally, 100%, stats done, HP calculated, treasure ready yadda yadda yadda.
Inspiration is the solution to most problems in gaming, but I don't expect the players to have it. They don't know the story and we (as GM/DM) don't always tell it the best way. I'm sure most of us have been in a situation where you acted in one way because you misunderstood something. Sometimes that is the fault of the GM, and they should allow a do-over, and sometimes it is the players fault, and they should be inspired to pay more attention.
Anyway, I'm ranting now. I'll stop.