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.