We run ours exclusively on 12v when on the road. Did it for 8 years before adding auxiliary charge line and ignition controlled relay to shut off refer with truck. Running out immediately when stopping for 5 min to keep batteries from discharging got old. Also shutting down refer last hour or to ensure batteries charged when arriving at camp.
Adding the ignition controlled relay simplifies usage on 12v, running auxiliary charge line makes it more practical as far as battery charging and refer 12v use. We also now have solar-running the refer on 12v just is not an issue. Running on 12v is the least efficient but it does work. That said as MOST will tell you you cant or shouldn't so I would be hesitant to recommend. Why it worked for us and seemingly no one else, especially for the 8 years prior to upgrades, I cant say.
Couple of thoughts-
Verified you have sufficient 12v supply to run refer when truck running? Looking back it appears you've added 10ga charge line. Have you tried to run the refer on 12v while traveling and can truck keep up? No sense going to effort of relay if it kills your batteries and milk is still warm?
Second is you said "sparingly for short runs not for continuous use".. If thats the case, the refer pre-cooled & closed will keep cool for a few hours with out running at all.
Third have you verified that refer will respond as you expect. IE set to run 12v, once on- disconnect the 12v source, pause then reengage-Does refer come back on in 12v mode. Some when source power removed then reapplied the board will default to off position. (so running a relay wont work).
IF you decide to add the ignition controlled relay, Id recommend adding manual switch to relays ignition trigger wire. Also, though it is as simple as adding a ign. controlled relay, you'll also need to run a 12v source selector switch, either directly to battery as normal for propane use or thru the relay. Could be as simple as an on/off switch (30amp toggle) on source line with the relay as a bypass.