Bride and bridesmaids getting ready together for the wedding is the one absolute thing I would expect of being a bridesmaid other than "attend the wedding ceremony." Honestly, all sorts of other stuff may be different -- I've picked out my own dress or had every detail dictated, I've hosted and not hosted bridal showers and bachelorette parties, I've helped with DIY crafts or just showed up -- but the one common thing in every wedding I've ever been part of is that the bridal party gets ready together. I'd never call a bride even a little rude for expecting that, and I'd go through a lot more inconvenience than figuring out an alternate ride in order to do so.
I don't see why it would be such a bad thing for your dad to pick you up. For one thing, the father of the bride isn't involved in "the girls getting ready," so he wouldn't be missing anything if he wasn't there. For another, he'd be bringing you for that part, so he wouldn't be missing it, he'd be arriving just in time. But there are other options too, if you can't get a ride there and have to drive yourself. If your house is only 15 minutes from the church, why can't someone else swing by to pick up DH and the boys on the way to the wedding? I'm sure out of all your relatives who are attending, someone can manage to pick them up. Even if it has to be your car for car seat reasons, couldn't you leave a little earlier than the other bridesmaids to pick them up? Or someone else leaving from the bride's house (MIL, FIL, one of the other bridesmaids' husbands/SOs?) could drive your car to pick them up, then you and your DH take it home. It seems like a pretty easy problem to work out.