I think simply killing off Bazaine is, well, lazy alternate history, so instead let's say that Regnier, after being unable to get an audience with Empress Eugénie, is unable to get his IOTL audience with the Prime Imperial, and thus never receiving a signed photograph of Napoléon III to use as credentials to get through the Prussian lines or to get to Bazaine, and therefore Bourbaki never leaves. However the old Marshal is still who is his; he rejects the Government of National Defense and what it stands for, and he believes that the war was lost before it began. However ITTL with Bourbaki still in Metz when Bazaine proposes to surrender Bourbaki refuses, and stages a mutiny inside the fortress which sees Bazaine arrested.
Therefor the siege holds on for another few weeks, which has some fairly drastic consequences. In the immediate period this means that Prince Friedrich Karl's Second Army never moves south, giving the Armée de la Loire time to regroup and organize after Coulmiers. As well the French Northern Army under Forre isn't forced to fall back as, with Metz still standing, for the moment, the French front isn't in a such an unstable position. So essentially the French still have the majority of their forces in the war; bloodied, but still fighting.
At the same time that Metz does fall in late November, the Loire Army begins moving north to break the siege of Paris, and while the Prussians begin to move out from Metz attempting to head off Chanzy the North Army commences a withdrawal just ahead of the Prussian advance, harassing and slowing down the Prussian forces, giving Chanzy the time he needs to reach the city. von Blumenthal knows he can't besiege Paris and hold off the Army of the Lorie at the same time, and, regretfully, detaches from the city, falling back to meet up with the approaching forces; Forre is smashed between the two of them, but his sacrifice allows the French to retake their capital and begin to reorganize. In Paris Chanzy, working with Trochu, d'Aurelle, Crouzat, and others, sets out to prepare their defenses.
Meanwhile, the Prussians, now united outside of Paris, are beginning to fracture as dissenting opinions rise up. von Moltke, as in IOTL, is worried that insufficient supplies are reaching the Prussian front as winter sets in, and is also concerned about the spread of disease among the German armies. von Blumenthal, who had commanded the siege, is also concerned that the Prussians now have insufficient forces to take the city, and advise attempting to draw the French out and defeat them in the field. Other German commands feel likewise; Frederick Francis II and von Werder being two other prominent voices calling for a change in strategy. However Bismarck, supported by King Wilhelm and the Crown Prince Frederick, as well as the Prussian Prince Friedrich Karl and Saxon Crown Prince Albert, call for an immediate investment in the city. They believe that holding back would only give the French further hope for a victory, especially as the Government of National Defense has already summoned every available French unit across the continent to the defense of Paris, even summoning units stationed in Algeria & Rome. Waiting would also allow the French inside Paris time to consolidate their position, and possibly allow Gambetta, who is in the unoccupied south, to raise a further fighting force. As well it could potentially allow the French to even turn diplomatic opinion in their favor and bring a third party (e.g. Austria) into the conflict. Bismarck in particular is also concerned about the affect a long, drawn out war, would have on the small and fragile Prussian economy, as well as Berlin's control over the newly-constituted North German Confederation. They call for an immediate artillery assault on the city followed up by a direct attack. Regretfully, Moltke gives the order.
The (Second) Battle of Paris is the largest single battle in European history since the Battle of the Nations, and pits some 650,000 French forces against some 575,000 Prussian and allied German forces. The French are behind significant defenses (which happen to be their capital), and have have better rifles (in terms of range), but are under-supplied, and half their forces are irregular and volunteer units. The Germans have superior rifles (in terms of rate of fire) and artillery, but are attacking a fortress city that can potentially hold out for several months, and if the Prussians don't take it immediately they'll soon find themselves under-supplied, deep in hostile territory at the end of a long supply train. For both sides this is the make-or-break moment.