Um...hmmmm...NO!
First of all, lose some weight. I don't think sport bikes carry guys who are over 300 lbs. My bike had a weight limit of 300.
Secondly, the GSX-R 750 is a very aggressive sportbike. It's not for the beginner. Typically, I would tell a beginner to start no higher than a 500 cc bike.
For someone your size, you really should consider either losing a lot of weight (about 50 + pounds) before getting on a sport bike, or consider a larger cruiser - like a Harley or a V-Star product - they may be able to handle the weight.
You're definitely not riding with your GF on the back of the bike - the bike simply won't take it!
Good luck, be smart, and diet!