Re: Shi@t test... or NOT
I would not push too hard to meet a girl you do not know well on her birthday.
The reason is that it is one day where she is highly likely to already have big plans. If you push to meet her and she first just ignores the suggestion, so you push harder and she has to turn you down she may then rationalise this to herself as, "having to turn down the pushy guy".
I would suggest being cool, and not bringing it up again. Let her have her birthday and then arrange something for a day or two later as if it is no big deal. If she does decide to invite you to do something, enjoy it! get her a little present if you see her, but make it more of a joke at this point than a grand gesture, which would in my opinion be showing too much investment too soon.