Woodstove Install in a Barn

Regardless of what you prefer, you need to meet both the requirements of your town/city/county (LAHJ) for proper woodstove installation, and your insurance company's standards, unless you operate "at risk" without insurance.

Both of those entities will almost certainly require transitioning to UL Listed Class A insulated chimney before leaving the room. If the beams in question are at, near (and you transition to class A before passing them), or in the ceiling space that will likely also address protecting them adequately, since those systems have much smaller clearances to combustibles than uninsulated single-wall stovepipe.

You will need to extend the exterior chimney at least 2 feet above the peak of the roof (2 feet above the highest part of the roof within 10 feet is the code language, so far as I recall at present.) You might be able to save money on Class A Chimney by locating the class A chimney at the peak of the roof (where it only has to stick up 2 feet above the peak) and angling the singlewall pipe to the center in the room below. That position is also less prone to leaks than a chimney sticking out 10 feet down the slope of the roof.

You will need to give consideration to access for cleaning your stovepipe and chimney, as well. A long run of singlewall can help to heat the space, but it also serves as a great place to collect creosote and start chimney fires.

It may also be required (and would be advisable regardless of that) to install smoke and carbon monoxide detectors in the space.