You're making this too complicated.
The fraternity isn't the be all and end all of life that you're making the big/little thing out to be.
DubaiSis has it right. Be direct and to the point. You like her - so tell her so and ask if you can take her on a date. You know, "Let's hang out sometime" is ambiguous given the big/little thing. So, make sure you actually ask for a "date" so that she's clear on what you mean. Be up front and acknowledge/discuss the fraternity situation with her if you have to go there.
Good luck.
