If I were you, I'd go with the coffee option because it's friendly and engaging without making you seem too desperate to start a relationship. Phone number and/or dinner both imply a commitment and you don't know what she's looking for yet. Better to test the waters by getting to know her first.
I get them as well, and they can be pretty painful. I find that if I press down on the area with my hand and inhale deeply for a few seconds, the bubble usually goes away.
Put a band-aid over it and tell her that you nicked yourself shaving.
If she's the type to ask to see the cut, then you might have to change your story.
In that case, scrape it up a bit more so it looks less like a hickey then mumble something about messing around with powertools. Good luck!