KCvale
In memory of KCvale 1959 - 2019
If I may make a recommendation...new wheel? maybe yes. a sun rhyno on a decent hub w better than stock right. or get a rim w the same inner rim size. disc adapter...
Just replace the wheel with an internally geared hub so you can use a nice HD 410 or even 415 drive chain that doesn't have to move from sprocket to sprocket laterally tearing itself up at least a little bit on every shift.
Heck you can get a new 26" coaster brake 3-speed Shimano back wheel with the cable and shifter for like $50 and though the coaster brake is useless with a jackshaft you have your bikes dual V brakes which is why for many shifter builds especially 4-strokers I go with a steel 7-speed bike and swap the hub or whole wheel for an internal.
Your bike is a fix but I routinely have the local bike shop I get my new bikes from let me swap out a 7 speeds whole system for an internal 3-speed for like $30 if I pull all the 7-speed stuff off and install the new wheel and shifter.
I have the parts for my next Big Red 4-stroke 3-speed shifter sitting here waiting for it's turn in the build bay.
Other than NuVinci's I have yet to see a trashed internal shifter and they are a joy to ride ;-}