No, I would not divulge either of my brothers number to anyone, without their permission. If it was a girl that was involved with me in anyway I'd steer my brother away from her because I care about my family and I know who she is... My little brother has offered up girls to me before that he showed up with (as a first date) and didn't really connect with, that showed strong attraction to me. I've always strongly refused in those situations. I would never date a girl that would use my brother for gain... How you treat someone's family is important. If she had no involvement with me, and she just liked my brother and was afraid to talk to him I might mention her if I thought she was a decent person from first judgement, but I wouldn't try to play match maker.
Honestly I would not give my brothers number right away. Firstly I would have to know this girl and like her. Secondly, I would ask my brother first what he thought of her. Thirdly, (If my brother is interested) I would give her number to him then he can decide what he wants to do with it.
Seems like you should give the number, your time is better spent on that which has true love for you, your dog. Dogs love you because you feed them and you love them because they love you, it don't get more honest than that. Trust your dog, have your affairs and when the affair ends, hug your dog, you will feel OK about it all
I don't hook people up. But I would try to get them together so they could maybe see if they actually do have a spark or not. She might just think he's cute and find out he's a jerk or just not her type. I would definitely have her figure out how to get it herself but put them in situations where it's an option.
TBH I would talk to my brother first to see if he would like to get to know her or if he likes her before giving her hope by just giving her his number. And I don't think your brother would like the fact that you just gave some girl his number.