A sterling engine is simply a gas expansion (or compression) engine that can convert heat to to mechanical work via air expansion in the the engine. So does a sterling engine ( or beta type in this case) have the capability to power a conveyor belt?
It does not expand or compress, it does both alternately.
So the gas (working fluid) wants to expand and contract alternatively ,when it is alternately exposed to heating and cooling. This alternating gas pressure can do work on a piston on both the upstroke and the down-stroke, it cases, that function is split between two pistons, but we lose the point. The point is alternating gas pressure can be harnessed for work, and flywheels, levers and such, tools at our disposal, make us able to use any power source for any task.
standard-wise this is not applied. You know all the power generation stirling generators. Making el. power and drive the el. motor of a conveyor belt is perhaps not what you want. I would try this site (many hobby guys, but also real experts)
Stirlingengineforum.com
I think the challenge is to find a company to build one or a very few customized machines for an acceptible price. You can not buy them "over the counter".
it is based on many parameters such as belt conveyer friction,.... so you should first design belt conveyer and after it design a proper engine for it.