Is there an instrument to measure/uniquely identify gadget's RF signal strength
You have already identified that the crucial part of this instrument is to only measure the signal strength of the voting system base station. If that unit used a standard protocol such as WiFi there are a number of instruments available that can do exactly that. Even an Android phone can run a WiFi Analyser app to give you an uncalibrated measurement of signal strength. But it sounds like your voting system uses a non-standard protocol. To be able to measure its signal strength you need to know:


the frequencies or channels in use
the modulation scheme
the format of the data packets so that you can decode the packet and identify that it is from the voting system base station.


The only people who will know all of those things are the manufacturers of the voting system. It is possible that they already have a signal strength meter for use with their system. I would contact them.