You do have to pop the buttons off gently - one at a time. Watch how they come off and organize a storage method so you get them back on correctly.
Having said that, and having completely disassembled my 1987 300D ACC, you may want to stop taking the thing apart right where you are now - and just resolder the joints you can see. What you see right now is the vast majority of solder joints. Its just the one circuit board.
What are the symptoms of the problem you are having? Mine was that the AC compressor came on in ECONOMY mode, and the fan would still run in OFF mode. I could actually see a bad solder joint in the circuit board you show. It could have been fixed with only taking off the bottom cover. So don't tear the thing down any further than you have too......