I have a propane fueled Hardwick stove. The pilot light on the oven lights and the secondary flame for the thermo couple lights. Once the thermo couple heats up the oven lights but after about 20 seconds the thermo couple flame goes out and the oven flame gradually goes out. It appears that the gas is being shut off to the oven flame. I made sure the propane tank is full.
Hold the torch to the pilot assembly and see if the flame stays on. If it does then repositioning the copper/nickel probe on the pilot light may make it stay lit. For optimum performance the flame needs to hit the very tip of the bulb. The copper/nickel bulb when heated produces electrical current which opens a bi-metal flap inside the gas valve. You may just have to replace that safety valve if that flame will not hold. Some times buildup on the pilot itself will make the flame smaller. Don’t try to open the pilot hole but instead try and clean it with a wire brush or some rustbuster.