Remove old non-indexed levers for gear changing to make "fixed" speed bike? I recently got an old bicycle for the sole purpose of using it on a trainer. It has old non-indexed levers for gear changing and they don't seem to work OK. The front one slips into lower after a while, and the rear one is now "locked" since it's plastic, I don't want to force it and break it.

I don't want to invest much into this bike, it was already more than I was planning to spend on it.

I was thinking to remove them and fix the derailleurs on a specific gear ratio. I would like to be able to adjust the rear one, but wouldn't mind if the process is somewhat cumbersome (ie: requires tools).

Is this possible? What would be the best path to do it?

Some pictures for clarification:

Shifters:


Front derailleur:


Rear derailleur:


There doesn't seem to be any barrel adjuster on the derailleur side, just on the shifters.
If I understand correctly you have slipping shifting levers that let the derailleurs move without your intervening. A non-indexed shifter slips because the friction setting is too low.

Friction of the lever is easily adjusted by a screw at the side of the lever. If it still slips after tightening you'll unscrew the thing taking care to remember the correct order of items when is comes apart. (Take a photo maybe to assist your memory.) Then clean the bits to remove any grease and reassemble. It usually helps. At least it did in the old days before indexed gears.