Dallas is correct, the water will flow through the pumps at a reduced rate if they are not running, Unless the pump has a flow control valve in it designed to stop 'gravity flow'.
If the PRV is located after the coil, and the cold water supply [well or water system] is before the coil, if the PRV opens, to vent a steam buildup in the coil, then the pressure of the water will force water through the pipes and the coil till the PRV has cooled off and closes.. So if the water stagnates in the coil, and boils, then the PRV will open, then close, again and again, the frequency will depend on the heat in the stove,and the temperature of the water in the piping. This is one very good reason to put a pipe, elbow and long 'drip leg' on the PRV and put a bucket under the drip leg. The steam and hot water will be caught in the bucket, and you can see what is going on, istead of water all over the floor.
Most circuits will thermosiphon some once they have been 'started' with a pump, but they will be pretty slow with the flow restrictions created by lots of elbow, the pump, etc.
As for using an inverter to run a pump. the small cartridge pumps only pull a little less than an amp.. the taco 007 is 1/25 hp, and pulls .78 amps.. virtually any inverter will handle this.. Unless you are using one of the 'oldfashioned' B&G model 100 pumps with the 1/3hp motor, they eat electricity and maintenance time and parts.. the cartidge pumps will run 24/7 for a decade or more,, I have several doing just that. The B&G pumps were a constant source of work and maintenance in my previous homes and heat systems.