If you say that the tank lid is covered, I'd doubt that you are getting water in the fuel tank through the tank bag.
Something to try in a low-light(dark) spot, look for stray "spark" around the spark plug wiring while the bike is running. Where the plug wires run along the edge of the head and around the coil packs.
If you don't see anything, then try a "spray bottle" of water and carefully spray some on the plug wires, coils with the bike running and see what happens.
Something else, on a weekend, or day off, get the garden hose out and wet down a specific area at a time as see if you can determine if the rough running happens when a certain area gets wet when cold.