The manual is right, but the pulling is in the down not out direction. I have removed them in both of my MH's and this is how I do it.
Open the window, slide screen to the middle of the opening, Grasp both sides of the screen and pull down to flex the screen more in the direction it is already curved, tilt/pull to get the top to come out and lift it out of the track. Reinstall the bottom first and flex it back in.
If replacing the actual screen material is what you want to do, you may be able to replace it without removal. If access to the backside is necessary, you should be able to get to it from the outside with the window open, I think...... I don't know if the center stiles overlap far enough to get to the spline.