Our Murphy bed mattress was/is horrible but we added an 1-1/2 inch foam topper and solved the problem pretty well. The construction of our trailer/Murphy bed would not allow any more thickness and still fold up and down correctly. We find the straps are necessary to hold the mattress and topper from sliding down while folded away. Even with the straps they slide a little while traveling and can be a PITA trying to pull the mattress back to the foot of the bed while it is in the air over your head.
Perhaps a custom mattress construction shop could be of some help??