Hi Daniel,
This is why you should always have current photos on your dating profile. It saves you the trouble of having to deal with issues like this. It may seem trivial at face value, but it isn’t. She started talking to you on the dating site based on what she saw on your dating profile.
She saw a guy who has no facial hair. That may be what she is into. Now, you could be wasting her time by meeting up if facial hair isn’t something she finds attractive on men. Even though it is just a stache and not full-on facial hair, she may still be turned off by it.
If keeping the stache isn’t a big deal to you, shave it off.
Don’t worry, if the both of you hit it off, you will be able to grow it back in the future. By that point, she would be emotionally invested in you, which would allow you to get away with certain things that she otherwise wouldn’t have forgiven if there was no emotional investment.
As of now, you are yet to reach this point of privilege.
Again, if the stache isn’t a big deal to you, just shave it for now. Keeping it on and sending her a photo instead could turn her off.
Even if seeing a photo of you with a stache doesn’t necessarily turn her off, she may still be turned off by the fact that you didn’t use current photos on your dating profile. That alone could put a sour taste in her mouth.
So far, the both of you have really struck it off and it wouldn’t be worth it to risk all of that goodwill by taking a chance on how she may or may not react to a photo of you with a stache.
Shave the stache for now and avoid any unnecessary and possibly unappealing reactions.