If Burnside attacks on the 14th (which as you say he DID consider, so assume he does for the ATL) and loses, does that significantly harm the Union, or was the ANV in such dire straits that a new attack could only be repelled with relatively acceptable loss-ratios for the Union?
Then Burnside would likely have been one of the many more corpses to be added to battlefield. He planned to lead the corps into the battle, straight up the center of the wall, it wouldn't have ended well to say the least.
Meade for example busted a hole in Jackson's line and asked Gibbon for support only for Gibbon to tell him off. If Gibbon had said great, I'm behind you, he could have widened the gap and still been following orders.
He found a hundred yard gap between Jackson's lines, could Gibbon have widened the gap? Of course, would that have meant a victory? No.
Regardless, a renewed offensive on the 14th with the Fresh Forces Burnside had would have crumbled Lee's Army which was spent from the fight and had no reserves. IX, XI, and XII Corps were available to renew the assault and if they cracked a hole, Lee's forces would not have been able to contain it.
Considering that Burnside wanted to attack the center of wall, it would be very unlikely that it would have worked due to a number of various things. Firstly, they couldn't outflank it from when they marched towards it, they could come along a small ridge that offered protection and keep going right but then they were met with impassible terrain. To the left you had not only no protection, but you would run directly into the wall if you tried to follow the protection.
Commanders, lost control of their men almost instantly once they were actually able to fire at the wall, the sheer amount of smoke, noise etc. made it nearly impossible, not to mention, you had a number of men that held the ground right behind the end of the protective ridge, these men, more often then not, fired into their own lines because of over-lap.
The best thing that Burnside could have done, would have been to go eight or ten miles upstream and cross at United States Ford ( I'm saying this all off the top of my head but I'm sure TFSmith will correct me if I'm wrong) Burnside decided to do just this during his mud march, but at that point, the rebels had fortified the crossing and the weather made the march grind to a halt.