Maybe. But perception is reality to a point. On the other side of the coin.....If a truly 'undecided' voter is turned off by 'transcript and birther stories' then this becomes a personal attack on the President. Whether valid or not, these stories will be interpreted by some as 'cheap politics.' Obama wins....and the more the 'free story' gets mentioned.....the more Obama is just 'being mistreated and unfairly characterized.' Just depends on the individual's take of the request/proposal....and that can be completely independent of Obama's response(s).
Donald's clearly not dumb...but he's brash. Romney has been gaining footing, in some respects, because of his 'cool' Presidential demeanor. He refused to wade into the mud in the last debate. Trump brings the mud right up to the steps of the White House. In my opinion, Obama has enough policy failures to focus on and Trump, in this instance, is distracting from that.