Is it possible to run it so it never heats up and only activates the neon? Would this be bad for the lamp?

When only the neon being ionised, the borate glass absorbing the argon in it, so the lamp will fail to start if all the argon from the penning mixture has been consumed.
Also, burning on the starting gas only shorts the lamp life.

I would suspect too cold operating electrodes, so way faster electrode degradation (sputtering).

When the SOX lamp lights only on the argon-neon mix, the borate glass absorbs the argon at its fastest rate, so the lamp may fail before the emissive coating on the electrode has been cosnumed.

The CFL high frequency ballast with it's high frequency and rather high voltage will ignite even quite deep "cleaned out" lamp (of course, on the rated gear it may fail to strike soon).

As an experiment, it will work for sure. Anyway, what are the other options, when such Neon-red lamp is what is needed?
Even when the SOX will last there way shorter than rated (for any reason), it still may be good enough for the task...

You can do it, with a high voltage high frequency power supply, and this will even light old EOL tubes but you might find arcing in the socket or inside the tube an issue if you go with to high of a voltage.. If you don't intend on using the light again at full power the argon clean up shouldn't affect it much and if you keep the current low like 1 or 2 watts you won't heat the sodium up enough to worry about it changing the color of the neon glow, but it might flicker in the dark which is a problem commonly found in older neon tubes, commonly like the NE2 indicator lights, they need a few photons to keep them glowing reliably, but just using a small blue LED is enough light if you can incorporate it in your design as a back-light to stop any flickering.  (this is done in modern neon ring counters to allow them to run 24/7, yes you can make a clock with nothing but Ne2 lights and it loses only 2 seconds a year, with out a GPS timing reference)