Hi Chad I assume you are installing individual rafters and tying them to a rafter beam on top.
Yes, the birdsmouth will slightly increase the angle from a pure 4/12 to perhaps a 4.1/12.....but why be concerned? (smile). And yes, the angle at the rafter beam will change ever so slightly....but remember, this is framing and not cabinet making (**smile**). I make sure the rafter beam is perfectly on center and is equidistant to the top plates all around, take the time to perfectly fit a rafter...then use it as a template to cut all others.
I usually limit my birdsmouts to less than 1/3 depth of the rafter, although you can use this depth to adjust each rafter to sit absolutely flush with its neighbor rafters.