I would like to know if I can use a variable autotransformer to lower the input voltage of a CWA ballast to dim the lamp it's running. I acknowledge that below a certain value, the lamp will refuse to run, and I would most likely need to start at the rated input voltage to let the ignitor do its job.

Yes this is technically possible. As long as you don't go so low that the lamp goes out, it should run. BUT:

HID lamps (particularly metal halide lamps) apparently like to get really fussy and upset if you don't strictly adhere to their rated current. Doing this long term will almost certainly reduce the lifespan of the lamp due to electrode sputtering.

Well I was only planning on doing it with HPS. And specifically with a cheap LEDvance HPS grow bulb, I wouldn't be too mad if it burnt out. Of course, I only plan on doing it for a short amount of time, just as a proof of concept.

Yea you are probably fine.

Not full range dimming but Advance used to specify alternative capacitor values and wiring diagrams to convert several models of CWA ballasts to HI-LOW bi-level operation. The document I attached lists several models of ballasts and the capacitors and wiring diagram needed to convert them to HI-LOW operation.

Some notes from the document:

• Hi-low dimming of many HID lamps is possible when using constant wattage autotransformer (CWA) ballasts by utilizing two different capacitance
values: one for full light output and another value for reduced light output. These capacitance values can be obtained using either a series or
parallel connection as shown below.

• The HID lamp manufacturers’ minimum allowable lamp power level is 50% of nominal lamp watts, in accordance with NEMA
Doc. LSD14-2002 (“Guidelines on the application of dimming to High Intensity Discharge lamps”)

• Always run lamps at 100% power (full light output) for 15 minutes before dimming.

I wonder why it was never noted that you need a really heavy duty relay or switch in such circuit or operate it very carefully at proper time, otherwise you end up with contacts welded, may be even on the first try

@switch rating: You need the switch to be rated for capacitive and/or inductive (depends whether you are switching coils or capacitors) load switching, that ensures all. But many people ignore those "nonsense letters cluttering the current specs", use the cheapest switching element with "the rated current" (so often meant for resistive loads only) and then are surprised...

I think I will run the bulb up for the first 15 minutes on 120v input on the ballast, than lower the input voltage by twisting the knob down on my variable autotransformer. That is how I'll dim it, but I want to know if there will be any effects on my variable autotransformer caused by the dimming process.

Almost certainly not, you will be reducing the power draw so it should actually be easier on your variable autotransformer. Probably.

OK good. My variable autotransformer is rated for 10A continuously so it should be more than enough for the task.