First, the Stirling engine is not a steam engine. It doesn’t boil water or use water vapor at all. It is an air engine, in that it works off the heating and cooling of air and the corresponding change in pressure or volume (depending on variant) that causes.
Second, Stirling engines, if properly designed, are usually more efficient than steam engines, that is they burn less fuel to do the same amount of work. This was especially true in the early days of both technologies, due to the rather astounding luck of the Rev. Robert Stirling in basing his designs on a closed cycle system that subsequent thermodynamic equations have confirmed as being probably as close to Carnot’s ideal heat engine as possible. This meant that his 1818 (or it could be 1827, I’ll have to check my notes) designs we’re achieving a 35-40% efficiency (that is, that percentage of the energy produced by the coal combustion was being successfully turned into useful work instead of waste heat) compared to the 3-6% of contemporary steam engines.
Third, Stirling engines have scaling issues. Stirling’s constant volume engines are constrained by an issue (I think it’s the heat exchanger, but don’t quote me on that) wherein the effectiveness decreases with size increases. The constant pressure engines that John Ericsson developed (yes, he of Monitor fame) got around this issue, but their power is constrained to their physical size; that is, making a more powerful engine requires linearly scaling up the size of the engine to match. Ericsson built an engine to power a paddle-wheel ship (modestly named Ericsson) in 1855. The pistons had a diameter of 14 feet, making them probably the largest pistons ever cast, yet the ship was still underpowered and probably would have required an even bigger engine had it not been sunk, raised, and outfitted with conventional steam engines that could deliver the required power without taking up all the space. This makes it unsuitable for heavy industry, however it is ideal for light cottage industry, where it is in fact used in many parts of the developing world today.
Fourth, everything above is good for the prompt, this isn’t. You really cannot develop a Stirling engine without tight manufacturing tolerances, ones that are non-existent prior to the 19th century. It works solely on tiny little voids between two surfaces and temperature changes are usually enough to make the metal expand and shrink a bit. Unless the designer and manufacturers know what they’re doing, it’s very easy to build an engine that seizes up or even tears itself apart.
So, economically viable, probably yes. Possible without earlier precision instruments? Not really, no.