You'll be fine with the Linbrooks.
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Since the room modes will be closer to each other in a small room, lets say every 50 or 60 hz, I think that you will have more acoustic problems. Also a 12 x 12 doesnt sound good, because its proportions are the same.
In a small room you will have more bass problems, and you will need to add some acoustic treatment to clean the soundstage and the image. Speaker and listening position experimenting will also help a lot.
My Linbrook System Is are in a 14X16 room with two openings (one to a hallway on one side behind the left speaker and the other to a small room on the other side behind the other speaker), and I would say they are large for *my* room (even though they sound fabulous despite this). Personally for a 12 X 12 room I would recommend the Linbrook System II (or even the Linbrook Signature Monitors). I have heard the System II, and it gives you a good deal of the sound of the I, while being a better match for the 12 X 12 room size. With your current room size the System I would be ideal.