I think there is a better solution..
Since a red dot sight generally has no power factor (ie it's a 1x), then you really only need the red dot, not the housing that goes around it.
I am right handed, but left eye dominant. This means I have to close my left eye to shoot, otherwise I confuse which bead to use causing crossfire.
Unit-dot
http://www.uni-dot.com/index.html makes a small aluinum tube that has a red, yellow or green fiberoptic rod in the back (they're interchangable). A special adhesive, which can be removed pretty easily, holds this on the rib just behind the bead. You cannot see the colored dot, which glows under daylight, unless you are aligned with it. Your shooting eye sees it, but your other eye doesn't. This allows both eyes to remain open, preventing crossfiring.
Now in your case, this provides a clean sight picture, with a red dot, where the usual bead is. You keep your head in the same position, and thus the gun feels normal and the sight picture is normal. And it is a lot cheaper than a red dot sight. In fact, the only drawback is that it is not as accurate as a red dot sight, but I doubt for birding and turkeys it makes that much of a difference. We're not talking about rifle accuracy. And a feature of the tube is that the eye must be aligned right or you can't see the dot. This helps folks with a consistent cheekweld.
Price is around $25 or so.
As for a red dot sight, my feeling is that it makes more sense on a slug gun, or on a dedicated turkey gun or barrel.