That's understandable. It's likely the white belt is feeling a little smug. He can go f*ck himself. That's his problem, not yours. If that entire school underestimates you, you'll murder them in a competition. Underestimating people is a common mistake. That having been said, I absolutely understand your desire to rep your school and yourself.
John Mejia has a good point about rolling hard in general. You need to train the way you compete. If the other person requests a light roll, by all means, oblige them...but otherwise, bring it. You can't always bring your A-game in your own school though. You need to develop your game and sometimes that means you will get caught by lower belts. That doesn't mean you are rolling light, it just means that you don't go straight for your bread and butter techniques.
So at another school, with people you haven't rolled with ever before - or in a long time, A-game is appropriate. Unless you have an itch to try something new - in which case, it's okay to get tapped.
John Mejia has a good point about rolling hard in general. You need to train the way you compete. If the other person requests a light roll, by all means, oblige them...but otherwise, bring it. You can't always bring your A-game in your own school though. You need to develop your game and sometimes that means you will get caught by lower belts. That doesn't mean you are rolling light, it just means that you don't go straight for your bread and butter techniques.
So at another school, with people you haven't rolled with ever before - or in a long time, A-game is appropriate. Unless you have an itch to try something new - in which case, it's okay to get tapped.