Hi, from my point of view, the main problem was the men in command, not the lack of pontoon bridges at time. Burnside had three "Grand Divisions" under him at Fredericksburg, each one commanded by a senior officer : Sumner, Hooker and Franklin. The "Reserve Grand Division", under Sigel, stayed at the rear and did not participate in the fight. The first one was actually the only one Burnside could trust to execute his orders, while the two others were not. Hooker, a competent officer, wanted to be in full command of the army while Franklin wasn't a bold fighter. Sigel wasn't at his best at the corps level, but he was beloved by his German troops of the 11th Corps.
The plan Burnside chose to manage was made for both trusty and able subordinates. Even if victory was uncertain, the actions would have been led with more efficiency (the advance of Meade's Pennsylvania Reserves showed great promises with the possibility of penetrate Jackson's Second Corps, forcing Lee to fall back).
Having the pontoon bridges earlier may have forced the Army of Northern Virginia to leave the field rather than fight and Burnside would not have scored any grand victory with this configuration of command. I think that even Sigel would have been more trustful than either Hooker or Franklin, despite his generally bad performance on the battlefield (Pea Ridge was an exception in his career).
Hooker should have been left at the rear (switching his role with Sigel) while Franklin should have been replaced by another officer (Couch seems to be a viable option, and his seniority was equal to Franklin's).
This means that, with the advantage of concentrating one month earlier than the historical Fredericksburg, Burnside wouldn't achieve any glory in the area because of this commanding system. He wasn't fit for that level of command, and two of his main subordinates were not willing to execute his orders, creating a lack of trust in the general command. An alternative Fredericksburg in this condition would have the same impact as Seven Pines (an inconclusive battle with no decisive effect on the military situation).
This reflects my personal opinion, I do not pretend to have the best answer available. Some other can consider the situation better or worst for the Army of the Potomac. I really enjoy this kind of discussion, this allows to debate on interesting military situations and lost opportunities, with the possibility of re-evaluating some of our favorite generals.
We can also imagine this scenario :
- November 17-18, Sumner arrives at Falmouth with the 2nd Corps (15,500 men) and the 9th Corps (16,500 men). He began the crossing of the Rappahannock and entrenches in Fredericksburg. The other federal units are on their way to reach him. Longstreet, at Culpeper, is ordered by Lee to delay the progression of Sumner. Jackson is urged to reach the Matapony in order to create a second line of defense. A.P. Hill's Light Division (11,500 men) stays on the Shenandoah Valley to threaten Washington and to prevent the concentration of all the federal army corps at Falmouth.
- November 23-25, Longstreet reaches the Marye's Heights, occupied by the 2nd Corps while Hooker and Franklin cross the Rappahannock. McLaws', Anderson's and Hood's Divisions (totalling 18,000 men) start to fire at the Federals in order to fix them while Pickett's and Ransom's Divisions (13,000 men) are sent on the federal left flank in order to repulse Sumner's Grand Division. This flanking column collides with some units of the 9th Corps and is finally repulsed after the arrival of federal units from the 3rd Corps (20,500 men). The 5th Corps (28,500), 1st Corps (16,000) and 6th Corps (24,000) are deploying on the heights, making any confederate assault suicidal. Burnside, having no information about Jackson's location, sent back the 11th Corps to Harpers Ferry to protect Washington against A.P. Hill's threat.
- November 29-30, Jackson's remaining units (15,000 men) are deploying along of the Matapony while D. H. Hill's Division (10,000 men) is sent on the south bank of the North Anna to prepare entrenchments, assuming that the Army of the Potomac will soon move against the confederate positions. Longstreet's Corps is retreating, having brought enough time to allow the concentration of the confederate troops down south. Hooker and Franklin, fearing too many casualties, do not pursue Longstreet, despite Burnside's orders to move forward.
- Etc. Winter Quarters, no major campaigns, except feints and skirmishes between the two armies south of the Rappahannock. The Federals establish a supply base at Fredericksburg while the Confederates develop their entrenchment system along the North Anna.