No. Not even a 2nd... The talent level from 6-14 or so is not that different. Not enough to move an early 2nd rounder just to move up 3 spots.
I'll have to disagree with you on this one. There is a fairly significant difference in talent and potential between the prospects that will be available at 6 and 14. At 6, you have the chance of getting Brayden Schenn, Magnus Paajarvi-Svensson, and Jared Cowen. At 14 you are likely to get a player such as Ryan Ellis, David Rundblad, Simon Depres, or Zack Kassian. There's a pretty large gap there in terms of current talent and potential. Maybe, in your personal opinion, you wouldn't trade a 2nd round pick to move up 3 spots, but it's been made pretty clear that that is what will be required to move up 3 spots. Last year the Maple Leafs traded the Islanders their 2nd and 3rd round picks in a package with their first rounder to move up 2 spots. The Predators made a similar deal with the Islanders to move up 2 spots as well. It also seems that there is an elite 3 (that will go in no specific order) that consists of John Tavares, Victor Hedman, and Matt Duchene. After these 3, many believe Evander Kane will go to Atlanta with the 4th pick. Then Schenn, Paajarvi-Svensson, and Cowen will round out the top 7. Picking tenth overall, you will be likely to land Nazem Kadri, Jordan Schroeder, or Scott Glennie. Although it is not overwhelming significant, there is a clear drop-off in talent and potential between these 3 prospects and Schenn, Paajarvi-Svensson, and Cowen. The difference between the 10th pick and the 7th pick is at least a second. It is also very conceivable to assume it would be a 2nd rounder plus a mediocre prospect or a 3rd rounder.
That being said, If Paajarvi is gone at 7, and the Leafs are in line to select Cowen, than i think Burke would consider trading down, and there is a small chance he would accept the OP's offer. I'm not sure how high the Leafs are on Cowen, maybe he's an ideal pick for them at 7.
Not really interested in taking on another project/gamble. We've already got plenty of those with guys like Grabovski etc. If he's got questionable work ethic/character then there's zero chance Burke will be interested.