Whether or not a 300/2.8 shot looks similar to a 200/2.8 or 300/4.0 shot will depend a lot on the ratio of the camera-to-subject distance to the camera-to-backdrop distance, and whether or not the scale of the backdrop matters. Perspective of a human subject doesn't change that much from 300mm to 200mm, because with both lenses your camera-to-subject distance is much greater than the size of your subject's head. As your lens focal length goes up, your "perspective" view of them asymptotically approaches orthogonal projection (perspective you'd have from infinitely far away), and by the time you're at the working distance of a 200mm lens, you're already most of the way there.
To get the same degree of background out-of-focus-ness of a 300/2.8 shot with a 300/4.0, what you need is for the backdrop to be farther from the subject than it would need to be for the 300/2.8 shot. This won't produce exactly the same end result, of course, since the background will be at a different scale. Whether this matters depends on the nature of the specific background and how much you're blurring it. With a 200/2.8, you will get decent background blur, but again the backdrop will look slightly less blurred overall because of the wider view angle of the lens showing you more of the backdrop (less "compression"). Neither substitute is going to give you exactly the same shot, it just depends on what characteristic of the 300/2.8 photos matter to you, and how much your shooting locations allow for getting pretty close to the same look.