Is there a way to play un-checked songs in a playlist?

OK, that makes it more complicated. I think, as you already surmised when you posted :-) , that there is not a way to do what you want only using checkmarks.

The trouble with checkmarks is that there is only one per song, and one of its purposes (skip when playing) may be at odds with another purpose (skip when syncing). You may have to bite the bullet and make a playlist for the car. Perhaps it can be populated by rules in a smart playlist; that will depend on what your library looks like. If not, you can let it be a regular playlist and just populate it manually.

Here is one way: Leave them all checked. Make a smart playlist called "Car" that includes the rule "Genre" "is not" "classical." Sync "Car" to your phone for road trips.