If Solo button is internal switch, does it sends any MIDI at all? Probably not...
Do bank buttons send On/Off MIDI? If yes, we can use them as "multi-purpose". F.e. when used alone they will switch bank. But if used together, we can have 2 mode switches: the left pressed when the right is hold and the right pressed when the left is hold. 2x LEDs, on these buttons can indicate the current status for both switches.
In fact I have already prepared a preset, with settings I have described before. But since we can not use "Solo shift" for modes, I can modify it to use something else.
I have started to write an "advanced" tutorial about Monitors. The text is lengthy, I have rewritten "theoretical" part 2 times already and I have a hope now that someone can understand it. I still have to finish detailed preset explanation before publishing the result. Compare to that, modification for mode switching will take almost no time