Changing a Jazzmaster rhythm circuit wiring

[Concretebadger]

I've noticed that the three-way toggle switch is on its way out so sometime in the next few weeks (probably the point where I lose patience with being unable to select the neck/bridge position) I'll dig the soldering iron out and replace it.

While the pickguard's off it would make sense to also remove the stock CIJ pots, file out the holes, fit some CTS replacements and generally tidy up the wiring before any of that breaks. Which leaves me with an interesting question: what to do about the rhythm circuit that I never use?

The pots attached to the roller knobs don't seem to suffer any pops or crackles so they're not really up for replacement. The JM rhythm circuit is infamous for being muddy and useless, but I think attaching it to the bridge rather than neck pickup might give some usable sounds since the bridge position has a bit more treble and 'spank' to begin with. Has anyone tried this before?