I think I have a solution to your quest that will work well without costing much money. All you need for this project is your propane burner, a pot, an immersion wort cooler, a temperature controller, such as an Ink Bird or a Baylite unit with a heating output plug, and a submersible water pump rated for high-temperature applications. You may have most of these gizmos laying around already. The pot can be your brew kettle or a smaller pot dedicated to mash heating, the immersion cooler can be used both for mash heating and wort cooling, and the temperature controller can be used for other brewing functions like keezer control. The only item that may fall into what Alton Brown of “Good Eats” calls a mono-tasker is the submersible water pump; the good news is that these little dudes are easy to find out on the interweb for about $50.
The basic setup uses your propane heater to heat a pot filled with water, the submersible pump to deliver hot water to the immersion coil (use high-temperature, braided hoses connected to the coil using hose clamps for safety), and the temperature controller to turn the pump on and off (pump must be plugged to the outlet designated for heating and the heating differential set to about 2 °F/1 °C). A good way to conserve water is to use the same water for mash heating and sparging. The water temperature is not critical if it’s about 10 °F (5 °C) hotter than your mash temperature set-point. You can either set your propane burner on the lowest fire to continuously heat the water or you can fire it on and off as needed.
In practice, consider using something like a metal grate to prevent the submersible from touching the bottom of the kettle and consider heating your water pot and immersion coil before mashing-in so that you don’t cool the mash with a cool coil. After your water and coil are hot, mix mash water and malt in your grain bag, let the mash sit for about 10 minutes to allow some mash thinning from enzyme activity, gently wiggle the immersion coil into the mash so that the coil is immersed, and drop the temperature sensor (make sure it is totally sealed and able to be dropped into liquid) into the mash.
Assume you start your mash at 149 °F (60 °C) and you have your controller set to 149 °F (60 °C) with the heating differential set at 2 °F (1 °C). When the measured mash temperature drops to 147 °F (55 °C), the pump will turn on and pump water through the immersion coil until the measured temperature is 149 °F (60 °C). Two practical problems with this design are short cycling of the pump and temperature stratification within the mash. The best way to address short cycling is to keep the differential setting to about 2 °F (1 °C) or greater. And the simplest way to deal with stratification is to gently stir the mash when the heating pump is turned on.
This basic setup keeps things simple without pumping wort or mash through a heater. It also closely mimics how commercial systems heated with steam operate. Hope this helps you get to where you want to go!