Use the coal fire as the high side and cool off the low side with your heating water. Seems that a differential of 2500 degrees far. should be a big enough difference. By the way, the pricetag on these engines are astronomical. Have you found a cheap source?
I found this on a stirling site.
"A good general guideline is that if the hot side of the engine is not at least 500 deg. F. (260 deg. C) the engine will be too bulky for the amount of power it puts out."
I think one could be built fairly cheap from scavenged parts. It isn't as complex as a steam or ICE
We need more info, plans and drawings.