Their website shows all the 5x12 coops have shed roofs. And raised floors. Is that what you have? Is the 4' you talk about from the raised floor? And is it to the top of the lower wall or to the upper wall? You might have options other than raising the roof straight up.
Aside from those questions, I think it is very doable. I added 2' sections of 4x4 to the tops of several of the 4x4 posts for my garden and it worked well. I used 6" (may be 5" or 7") corner straps - two for each post, on opposite corners of each post. Something like this picture - I remember we looked for ties that put the holes at different distances from the edge to spread out the strain. We drilled pilot holes and screwed them on. I was amazed at how stable they kept the joints over 5 years in the open; I probably shouldn't have been so amazed based on what they are designed to do. They didn't have much load, though, just fencing that I put up each spring and took down each fall.
If you put the new siding on so its joints are not in the same place as the joints of the posts, it will help stabilize the building. I think, if you set your saw to a shallow enough depth, you can cut the existing siding off while it is on the coop. I'd go 6" to 8" down. That lets you add a 2' strip (allowing for the open eaves at the top). It also lets you put the corner ties on the outside corners and directly on the existing posts (instead of through the existing siding).
I'd do the outside corners of the corner posts with the straps. For the inside of the corner posts and the studs, you could use scraps of wood to splice the joints.
There are other ways to do it, using timberframing methods, but they don't work enough better to justify the added skill and/or tools and time if you want the most practical route.