Cool idea! One of the AOS magazines a few months ago had an article about someone hanging large cork oak logs in their GH and covering them with mounted plants. It looked spectacular! Doing a mini version in your kitchen would be pretty awesome.
Not sure about the birch, I've never used it for mounting and don't know how well it holds up over time. I do know the black birch branches and logs I see fallen in woods around my area always seem fairly rotten, though I have no idea how long they've been sitting there... The birch could be pretty heavy, too, esp. for a log of that diameter.
Cork on PVC might work well and ought to be very durable. The big box pet stores in my area often carry cork tubes (for reptile use, I guess) that could be slid over a PVC pipe and fastened together.
Bear in mind that such an arrangement will likely be hard to take down and water, so you'd probably need to mist or water in place, which shouldn't be too much of an issue if it's over your sink. Dens and Phals also tend to have somewhat different cultural needs, so you might need to do a lot of spot watering and tailoring of micro-habitats with sphagnum moss and the like.
Can't argue with what Joann said about timing of mounting. I've always had better success mounting plants when their roots are in active growth, which can vary a lot from genus to genus. Might be a good project to build up and add things to over time. Keep us posted on your progress, it's a great idea!
--Nat