I think that a shade that picks up the colors in your hardwood floors would work well. However, you have a really wide window, so imagine that means a custom shade. I generally lean toward neutral toned curtains myself, but your room is all neutrals now. I think it could use at least a little color -- maybe even just some larger house plants would help.

Woven blinds would block some of your view even when fullly open. Since you have such a lovely tree to see, I would do just linen drapes. I watch Lone Fox videos on youtube, and he often uses this brand and highly recommends them.

Here’s a view of them in one of the rooms he designed.

I like your original idea of a woven shade with draperies, and it would look great with your furnishings. Just mount the shade higher than the window so when they're up you get the full view out your window. Yes, move your sofa a few inches off the wall so the draperies don't hit the sofa.

For that space I would inch the sofa far enough away from the windows to allow very easy movement of drapes, and I would think about doing 4 separate panels in case sometimes you only wanted to block some light.

Almost any type of quality drape would be lovely, light filtering or lush.