I'm no engineer either but I'm trying to think of the practical aspects of the installation. Good luck getting any manufacturer to sign off on it.
A masonry stove is great. You can't really burn a wood fire cool enough (or short enough duration) for heating our homes so like a
RMH you're trying to capture some of the heat lost up the flue. Again, it's to capture heat that would otherwise just be lost up the chimney. Modern gas burners are pretty darn efficient and there isn't that much heat lost. To make the thing work like a
wood stove the burner will run steady for .. an hour? 2? So you'll use more propane than you need.
How will you regulate temperature? Think where your thermostat is. Sensing room temp as a normal thermostat? The burner will run for a very long time before the room warms. Probably now the lag time has been so long you'll get a large overshoot on the temperature so it's too hot. So you'll use more propane than you need.
What if you just have a sensor on the stove itself? Might work but doesn't take into account the variations you get from a warm day or a cold one. Windy day or not. So you'll use more propane than you need. Or worse, some cold windy night a
water pipe in the corner will freeze even though the mass is at your chosen temperature.
A separate install with a normal room thermostat will heat the air just enough and then cycle off. It will take into account whether it's daytime and the sun is beaming in or a windy night, only using just enough gas to maintain room temp. The thermal mass of the stove will still do it's job, toning down those fluctuations in temperature.
What might be a good idea though is to have the stove close enough to a window that the sun can shine in and passively store some of that heat on a sunny day.
Here's the real bonus as I see it. Those long winter nights would be so much nicer if when the wood fire burned down in the middle of the night and you're faced with getting up in the dark to toss another log on you heard the propane burner kick on ... just roll over for another hour or two. It would heat the air just enough till you get up and stoke the fire again.
Could you run the exhaust pipe from a propane burner through the mass? Maybe but if it has to be inspected and approved.... good luck. Too much moisture in the gases to use the same flue as the wood burner I think.