It’s a pretty reasonable drive and driving will save you some trouble of shipping/boxing up/rebuilding bikes – unless of course you’re used to doing that and you sound like a family who rides.
It will also give you some time with your teenager and that can be rare. They are obviously interested in spending time with you if they suggested RAGBRAI to you.
Parking at the end town is never an issue and taking a charter that will ship your bikes and you across the state is pretty easy – I’ve used Brancel and thought everything about their service was great. Can’t remember pricing – not too expensive and not too cheap for a non regular route.
And as someone else pointed out getting home will feel that much faster!
Good luck and enjoy it. So many of us cannot wait to do this for a really really good reason!