Convert battery powered to battery backup

I just installed one of those cheapo battery powered (two 1.5V AA)
thermostats. It occurred to me that I could run a regulator from the
24V supply in the wall to derive the 3V needed to run the unit and use
the batteries only as a backup. What would I need to do to safely
wire this regulator in parallel with the batteries so that both can
supply power to the unit, but so that no current from the regulator
runs through the batteries? Any alternate takes on this idea?

.

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