A quick analysis of your issue is not so much long term compression, but how to quickly compress a sleeping bag and some SLD synthetic top and bottom quilts, to place into a backpack.

What if, you have an small electric pump attached to a vacuum bag with the items already sealed inside and the pump attached?

If you get the call, reach over the back seat and turn on the pump and drive, that might take 30-seconds. There are pumps that auto shut off when the max vacuum is achieved; so no worries there.

Upon arrival, go ahead and drop the vacuum sealed bag and quilts into the top of the backpack, and you are good to go.

Easy enough to do some practice runs to test.