22 ft unsupported seems like a very long span to me, what's that nearly 7 metres?
Sounds like it's getting into the realm of structural enginneering not diy for me.
If you want to save costs you might think aout a "flitch beam", that's 2 wood beams with a steel plate sandwiched in between - the three components are bolted together every few feet. Easier to join to the timbers then. https://www.youtube.com/watch?v=QWUNd559UQY
I still think you might be more like 10"x2 or even 12"x2 timbers to cover that span if totally unsupported. But might still come in a little cheaper than the i-beam.
Maybe the roof will be very lightweight and no snow weight is expected - but I'm no structural engineer so don't take my word for it.
Other features like corner bracing or canti-leverage, or some other support structure or other feature (like is it the bottom side of a framed gable triangle) might also help.
A flitch beam sounds interesting. The problem with 2-by timbers, like 10x2 or 12x2, is their thickness. A 12x2 would drop down 12 inches, and the highest point of my carport is only 8-feet high. In its 16-foot run, it has to drop at least 4 inches in order for water to run off. I can’t be using material as thick as 12x2. That’s why I was considering doubling or tripling some engineered wood. But at that point it seems like steel would be the same cost and better performing. I dunno. Like you said, I may have to talk to an engineer.