Switching based on voltage
That's a proximity sensor, not a &quot;probe&quot;.
Unfortunately there is no datasheet in the link you provided - just a basic specification so it may not be a good choice.

The probe's description says it supports DC 6 - 36 V, but I have found that I get the most consistent measurements at 12 V.

This may be true although it is unlikely. The sensor will (on a good one) regulate the voltage internally so that the sensitivity is independent of supply voltage.

I am looking at using a low-side transistor switch as a possible solution, but I am not sure.

A low-side transistor switch is not a voltage regulator. It's a switch.
Check your hunch about voltage sensitivity by mounting it in a way that you can make repeatable measurements. Be aware that their recommended method of operation is to have the target move across the face rather than towards the face. Edit your measurements into your question.