No, flip up bicycle seats are not common.
For (probably) less than $10US, you could get a quick-release seatpost collar so that it's easy to move the seat up or down. Just make sure to mark on the seat post what height you like it at.
For over $100US, you could get a "dropper" seatpost (intended for mountain biking) and set one of the lower positions to be the comfortable one. That would be a lot faster than quick-release option, and do much better at getting back to the same seat height setting, but it's a lot more expensive.
Alternately, you could look into other seat and/or rack options. A trailer made for hauling a kid? A baby seat that goes in front of the rider instead of behind? A longer rack? A rack that sits further back or up or down or something in a way that isn't such a pain? A cargo bike with a really long rack (so there's still room for panniers)?
