While could have ordered Sumner to capture the town of Fredericksburg itself, without a pontoon bridge Sumner cannot cross the river in large enough numbers to occupy Marye’s Heights, just the town itself. Not to mention the very legitimate fear that rising water levels would mean the isolation of Sumner's force. Sumner even later claimed that Burnside was right.
An easier POD would be having the pontoon bridges arrive at Fredericksburg on November 20-22, then Burnside should be able to capture Fredericksburg before any meaningful Confederate opposition be made to resist the crossings. IOTL, Lee anticipated the fall of Fredericksburg and had planned to concentrate the Army of Northern Virginia at North Anna. However, Burnside had successfully surprised Lee and thus the Army of Northern Virginia was still on the move. I believe Lee had intended to slow Burnside with an advance guard of McLaws' and Ransom's Divisions while the rest of Longstreet's and Jackson's Corps concentrated at North Anna.
Assuming the Army of the Potomac was fully concentrated at Fredericksburg by November 24, Longstreet should be in place at North Anna while Jackson is still in the process of marching to North Anna but is closing in. Burnside might be reluctant to attack Lee, due to a lack of knowledge of Jackson's location after crossing the river. Lee is in trouble; assuming no fighting breaks out between the advance guard and the Federals, Lee has only 38,320 men to cover the North Anna, far less than what he had during the Overland Campaign. Still, Lee was not the only one with severe trouble, Burnside was seriously struggling with keeping his men fed due to atrocious roads. So the way I see it:
- Burnside decides to establish winter camps at Fredericksburg and resume the campaign when May begins. This is fairly unlikely for Burnside. The man was pressured by Washington to win a major battle for the sake of the Emancipation Proclamation. Besides, it would seem like a waste to throw away the initiative after surprisingly Lee badly.
- Burnside attacks Lee's North Anna position. Lee's North Anna position is formidable, so long as Lee strongly contests the crossings, Burnside will have a hard fight securing a crossing. The main advantage Burnside can expect is his overwhelming numerical superiority over Longstreet. If Burnside breaks through one of the crossings, Longstreet will have a hard time containing the breach. However, Jackson's arrival makes a successful crossing a double-edged sword. If Lee throws a strong counterattack against one of Burnside's bridgeheads, it is probable that the bridgehead collapses with heavy losses. Whatever the result may be, Burnside must go into winter camps fairly soon. There was not much time to continue the campaign before winter made operations too difficult.