Face the fact, spur gears will be noisy, even in good condition, with correct backlash and accurately centre distanced. Otherwise helical gears would probably not been invented.
Listen to a Bedford OB or Austin K2 in the lower ratios to hear characteristic spur gear noise, or any spur gear transmission.. When automotive gearboxes started to use helical gears, before the days of synchromesh,, the selling point often was "Silent Third"
So, OP, resign yourself to some noise when using backgear or changewheels.
The smoother the drive input, the less noise will be evident Which is why 3 phase drive motors produce less noise than single phase
Back gear centre distances usually cannot be adjusted, but changewheels can, and should be.
By adjusting the banjo, and the position of the gears carried on it, the meshing can be improved.
Too little or too much backlash will increase noise, and probably, wear
The normal technique is to run a sheet of paper ( About 0.003" – 0.075 mm thick ) between the gear on the Leadscrew (Since that cannot be moved about ) and the Idler gear meshing with it. That will provide a little backlash
Then set the backlash between that Idler and the next gear in the train, and repeat for any others by positioning the studs on the Banjo.
To set the backlash on the final mesh between first Idler and the drive gear (Below the Tumbler reverse ) the Banjo will need to be rotated about the housing surrounding the Leadscrew..
Once all the backlashes have been optimised, (Check again after everything has been clamped in place ) noise should be reduced, but not probably entirely eliminated.
Sorry if this is teaching Granny to suck eggs.but if your changewheel drive is noisy, better setting of backlash ought to improve matters.
Howard