If it is definitely oxidation and not dust etc. getting into the contacts...
Going for minimal modifications, I wonder if a bit of DC (a volt or less maybe) through an RF choke on the switch's common terminal would give enough of a quick current spike to be useful. This depends on the following stages being AC-coupled up-front though (probably need a large resistor from each output terminal to gnd just to bleed off the series caps), and the front-end being able to tolerate a large-enough temporary spike to be useful. Or as an alternative, if the downstream stages have a DC path to ground but there's some capacitance on the common side of the switch, an RF choke/1M resistor can just charge up the common-side capacitance for a nice discharge when you rotate the switch.
Or since it sounds like the switch contacts are sitting out in the open, maybe there's a decent way to remove the switch without too much trouble, and electro-plate the contacts with gold?