re: help diagnosing electrical problem and possible solution

No, you'll have either something non-functional or you'll trip breakers if the two hot legs are on different sides (phases) coming from the circuit pannel. I just took a good look at your photos and I have a good understanding of what you are dealing with now, the problem is that you really need to trace that 12/3 wire and see where it goes. It's feeding two fixtures or a ceiling fan with a light, the black from the 12/3 is going to the wago connector and is providing the hot legs of those two left switches, the actual neutral wires are tied in at a junction box or the fixtures wherever they are. Unless you can get eyes on where that wire is going you're going to have nothing but guess work as there are a number of possibilities that I can think of in what you have to work with and it can become a very confusing mess really quick.