We have one of the sturdier beds from ikea with a different mattress but its great and very solid.
I used to have the one your looking at as a kids spare bed when nieces and nephews stayed and never had a problem with it. Perhaps if your concerned about the slats moving get DH to secure a couple with screws so they can't move? I would also try out some of the better mattresses as the cheapest one was not very comfy.
You should have no worries fitting the base in a standard car and I think some of the mattresses come rolled up so do some research before hiring the trailer.