I say stick to your guns- tell them you're not biking to the reunion so that they can bring another car. You ride a bike because you are trying to reduce The number of cars on the road, not to make it more convenient for someone else to drive. If they'll take umbrage to that, who cares. You can't tip-toe around people's feelings when you are doing something you believe in and that is worth doing. Family or not, nothing would ever get done if all we did was worry about what other people feel/think.
And yes, my acquaintances and family think I'm an ******* sometimes regarding things I believe in. But they get over it, and the ones that don't are not worth worrying about.