I just ran into this problem myself. The fix I've found is to setup the Radeon Pro executable itself as an application through Mod Organizer, the same as you did for TESV, SKSE, etc..
To be specific, you'll need to launch MO without having Radeon Pro already open. Once MO is active then launch Radeon Pro from the MO dropdown list. It should start up like normal, unlock MO and you can continue with whatever you were doing.
This process is a bit of a burden (at least for me), but it has been working consistently for getting my RP profile to execute with Skyrim through MO. Hope that helps.
Note: Do not run MO as administrator, otherwise Radeon Pro will not run and you'll have to relaunch MO (Win8, not sure about Win7 but I assume it's the same). Also, I've had problems with RP forcing the "failed to write log" error in MO after exiting Skyrim and trying to launch other MO-based applications. If this happens just exit Radeon Pro and relaunch when you're ready to go back in the game.