Nope, none whatsoever. Thats why it's there actually. I'll leave the headstock down on some machines if they primarily only use stubby 1/8" engraving bits, and raise it on another for use with endmill holders or rapid tool changers like you have. Pretty much it's there specifically for the kind of problem you are having, to ensure access to the full Z travel no matter what tooling you are using on it.
If you take the dovetail plate off, you will even notice that the tapped holes are already there to move the plate itself up or down and inch or so either way if you need even more clearance. You'll need to use an indicator and make sure you reattach it square if you move the plate itself, but there in no reason why you can't if you really need to. Remember the motor mount will have to come up the same amount to keep the belt straight in any case.