I'm going to have to double down and disagree with you yet again. "I have a boyfriend" IS the oldest trick in the book. Every girl that has ever been approached by a guy ever who was either out off by them or not convinced enough by them that they should take a risk on going out with them has used the "I have a boyfriend" excuse.
Did OP even ask to find out what the nature of this "boyfriend" is? How does OP know it's not just some guy she's been talking too but isn't that serious about. How does OP know that she's unhappy with her relationship with him and is looking to perhaps see someone else.
OP doesn't know any of that because all he probably did was walk up to her, chit chated for about 30 seconds asked her out without engaging her on an emotional level with out building up any kind of excitement or nervous/sexual tension. He might as well have dropped to one knee and begged her to go out with him.
Now I'm exaggerating a little bit here, I'm sure OP is a awesome guy with tons to offer women, but by merely propositioning a girl for a date without making her feel excited about that prospect by being light hearted yet persistent in your intentions your might as well asking HER to take YOU out on a date, and that's not how it works.
There are so many great "pick up" videos on YouTube that show guys steamrolling through that excuse like play-doh. Whatever OPs intentions with this girl even if it's romantic and not trying to sleep with her, he's got to learn how to approach girls with the outcome of what he wants in mind and SELL her on it. Sell her that going out for coffee with him will be the best decision she makes for what to do that day. Sell her that taking time to get to know him will be something that she'll want to do more than once. She's gotta have butterflies in her stomach thinking about that date.
Now I'll make a quick point here that some girls don't need more selling than others. Some just want to be approached by anyone because no one ever has the balls to do it and it will be easier for her to say yes, while others might be harder to convince because they have "options" so to speak whether they have other guys trying to court her or they just have other stuff they'd rather be doing.
Here's what I'm picturing in my mind that OP did. He goes up to a girl engages her in sort of a shy/reserved way, asks her how she's doing how her days going. His arms were probably folded up in his chest, he was probably a bit nervous and was half expecting that she would say no. If that's how he was, if wasn't already convinced in his mind that a date with him would be one of the best decisions shell make for what to do that day she's going to pick it up in his body language, she's going to pick it off listening to the volume level and intonation in his voice and she won't even be doing it consciously. On an sub conscious emotional level she'll feel OPs insecurity and it won't matter at ALL what's coming out of his mouth.
As far as how to deal with rejection. The best way to deal with it is to smile and convey understanding, shrug it off, laugh about it and know in your mind that there's still a million other women out there and that this is nothing but a minor set back. Don't end on a note of disappointment. Don't disengage the conversation with a sad pouty look on your face or try to leave her with guilt for rejecting you because you never know if she might change her mind and you want to end pleasantly friendly with a big smile on your face to keep the option open.
I think I'm 100% right on this, I feel like a lot of this is just common sense and I wasn't trying to make OP feel bad or make it seem like I think he's a pussy whatever. He had the balls to ask her out, I'm just saying that the proposition is not good enough you've got to try harder to win her over.