I also think your co-worker are jealous.

This being said, although their motives are silly (the grease thing), I have to say that lots of buildings don't want bikes in the offices. The main issue I've heard is liability in case of emergency. If you have a safe spot to put it in the garage, then good.

I think it's totally possible to bike to your meetings without sweating. Depending on the bike you ride, though, you might want to be careful about getting grease on your right pant leg. Rolling up said leg is probably the safest way to proceed if you don't have a chain protector (and even if you do...) You might want to do something about the left leg too if you're wearing really nice pants.

On a one-mile ride, you want to go as slow as possible, look at the birds and the flowers (or nice buildings), etc. to make sure you don't go too fast and sweat. Or if you are riding a speedy road bike and really can't help it, then you could consider getting a cheap city bike to keep in that same garage just to go to your meetings. When you switch bike, your body knows you're not in for the speed...

Good luck! I'm sure you can get rid of those excuses.