I'd take the lever out of the housing and spray it. No need to do a deep-dive disassembly, but I'd think you'd want more access for spray/runoff than you'd get leaving the lever in the housing.
I think you can take the black cover off without too much trouble, maybe just one small screw holding it in place? Might give you a better peek at the innards.
I'll take a closer look and see how daring I'm feeling! I do have a good set of allen keys, so the tools shouldn't be an issue.