Dimension an open-feeling deck roof

[Laxekongen]

Hello
I am planning to build a porch roof on the summer cottage this spring. It will be 5.4 m wide, attached to the house body, and with an overhang of 3.1 m out from the facade. Double-seamed sheet metal, paper, rough-hewn boards, 2.0 in snow load. We are going for the minimum slope for double seam, i.e., 6 degrees to minimize the loss of roof height at the front edge. 6 degrees result in a 32.5 cm loss of roof height from facade -> front edge. We have about 20 cm of eaves on the house and want to have about the same on the porch roof. The distance from the floor to the gutter is 2.85 m.
To achieve as open a feeling as possible on the porch, we want maximum height from floor -> support beam at the front edge. With 6 degrees, we have a height at the front edge of 2.85 m - 0.32 m roof slope 6 gr - dimensions of beams + rough-hewn boards (180mm + 34mm?) - gap gutter metal roof about 100 mm? - dim support beam at the front 2*45*120 = i.e., about 2.1 m in the front. That is a bit low, I think, and I would like to have at least 2.2 m. Can it be solved better to give more height?

Additionally, we want as few posts as possible and as much distance between them as possible on the outer support beam. We also do not want an extra support beam halfway out to avoid blocking the short sides with posts. How to solve this with suitable dimensions and what construction (attachment posts-beams-support beam) builds as little as possible?

Right now, I am considering 4 posts, i.e., 5 m - 2*0.2 m eaves / 3 => cc 1.67 between the posts and beams then at cc 83.5 cm. But can I reduce it to 3 or just 2 posts without the support beam at the front eating up the height? For example, as I have seen in many houses in the US, build on Y:s on the posts that offset the support beam => smaller cc dimension slimmer support beam.

I am toying with the idea of using glulam for the 4 roof trusses that lie directly over the posts and instead reducing the dimension of the 3 that lie between the posts and cross-bracing them against the stronger glulam beams. Then I can place the front support beam partly between the glulam trusses that the slimmer roof trusses sit on. In this way, I don't lose as much roof height for the support beam. Does this work?
Has anyone had the same dilemma and solved it wisely & cleverly?
Steel is always an option, but I suspect it would be too expensive.