I haven't looked at this unit in as much detail as my other lock-in amplifier, but I presume that the microcontroller is heavily involved in handling the reading of the buttons and the control of the indicator lights.
The problem really comes down to isolating where the faults are, one by one, and fixing them.
So far we have established that there aren't any obviously huge faults in the power supply. That's a good start.
Also, it seems that you can drop the fear that the high signal level caused this.
If the fault with the buttons turns out to be intermittent (i.e. it hasn't magically fixed itself) then I think this is the first problem to tackle. It might be a mechanical issue (e.g. a broken solder joint), a random piece of conductive rubbish causing occasional shorting, a failure causing occasional lockups of the processor, or unfortunately any one of many more things.
The first thing to do is to try to determine if there's a pattern. Does it always happen as you turn the unit on, or can it happen after a random delay? Does a bump or a jolt to the unit cause/fix the problem? Are any of the boards sensitive to pressure (carefully poking around with a plastic rod or similar).
In my repair of the other unit I have, I noted that while the findings were correctly selected, the lights would follow the sequence 1-2-3-3-1-2-3-3... Rather than 1-2-3-4-1-2-3-4... That pattern was the clue for me.
See if you can find something.
And, how long do you have to fix this? A few days? A few weeks? (How long until you need to make a final decision?)