The bed seems to be working nicely!
Regarding your HVAC installation, you may have already done this but if not you'll need to supply outside air to the intake vents on the sides and top of the unit.
The thermostat can be relocated. But before you try that, I'd make a deflector to send the cooled air upward. I have a feeling that it is just "looping" back to the return air grill.
If the deflector doesn't cure the situation, then I'd have a go at moving the thermostat. Check the wiring diagram - either in your manual or inside of the case of the window unit - to see how the compressor and fan motor are controlled. If relocating the thermostat requires moving any of the 120 volt A-C wiring, then be sure to discharge the capacitor before doing so. It looks like the membrane button controls are part of the front cover. If so you may have to cut out that portion of the plastic from the front cover.
I'm in the middle of an HVAC install in my teardrop. As part of my "hack" of a window unit, I am removing the controls that came with the unit and changing over to a residential style thermostat. This too is a possibility but is more involved than using the factory thermostat. This thread shows different "hacks" of window units:
http://www.tnttt.com/viewtopic.php?f=30&t=44561You may want to peruse that to see what is involved on modifying a window unit.