The battle of Mars-la-Tour was the greatest opportunity to inflict a serious setback upon the Prussians that the French missed.
After the French suffered several defeats at the hands of the Prussians. Napoleon III and the Army of the Rhine retreated to Metz. There, the Emperor gave the command of the Army of the Rhine to Marshal Bazaine, ordering him to bring back it to Chalons. But Bazaine wanted to deliver a battle under the walls of Metz, thinking to be able to smash the ennemy here, and delayed the retreat. Thus, he accepted to deliver battle at Borny while the Prussians crossed the Moselle south of Metz. The Prussian vanguard, von Alvensleben's 3rd corps, skirmished with French detachments near Mars-la-Tour in the night of the 15 to 16 August (it was during this very night that Napoleon III and Prince Napoleon left Metz, passing close to Prussian outposts). Von Alvensleben believed then to have in front of him a rearguard, believing the main French army to be heading for Verdun, more at west, and decided to engage. However, the Army of the Rhine, delayed by Bazaine, was still near Metz. Thus, the French brought their entire army on the battlefield by the afternoon of August 16th, while the main Prussian forces weren't to arrive before the following night.
From what I've found, the best occasion of the day to have a great Prussian defeat was with the fights of the 'Ravine of the pears', in the western part of the battlefield.
At 5.00 pm, von Schwartzkoppen's 38th brigade launched an attack on which was believed to be the right flank of the French, in fact the left flank of de Ladmirault's 4th corps. After having crossed a plateau, the brigade had to cross a ravine before arriving on the French positions.
Once at the bottom of the 'ravine of the pears' as it was called at the time, they took heavy losses but were about to overrun the French position when de Cissey's corps arrived, reversed the situation and routed the 38th brigade.
The Prussian cavalry, already strained by important fights more to east, attempted two counter-attacks which both failed.
At this moment, the 38th didn't exist and nothing stood between the French 4th corps and Tronville. However, de Ladmirault halted the pursuit, believing to face the Prussian 3rd Army (which was in fact near Bar-le-Duc); it's surprising for such an agressive general (relatively to the other generals and marshals of Napoleon III).
Had he pursued, the von Alvensleben's 3rd corps could have been surrounded.
My main objective by making this thread is exploring the ways possible to have the Army of the Rhine make junction with the army gathering at Chalons, preventing the disaster of Sedan.
Even if we can't make the French win this battle, we can eliminate Bazaine. Indeed, at the beginning of the afternoon Marshal Bazaine was surprised by a charge of Prussian hussars and escaped in extremis to capture. Earlier, he also had his horse shot under him.
With Bazaine removed, Marshal Canrobert would have taken command. He would have two choices for the following day:
> he begins immediatly the retreat towards Verdun;
> he reorganizes after the removal of Bazaine, waits for the reaction of the Emperor to head for Verdun and uses the following day to entrench, but on an west-east axis to keep open the road to Verdun instead of the north-south axis chosen by Bazaine (who wanted to deliver a great defensive battle under the walls of Metz, not to head for Chalons).
If the Army of the Rhine and its 150,000 men (once a garrison of 20,000/30,000 men let in Metz) successfully make their way to Chalons, Mac Mahon could retreat to Paris as originally planned, with 250,000 men, enough to prevent a siege. By autumn, with general mobilization decreed and completed, the Germans should face around 1,400,000 French soldiers. Given the difficulties they had with the siege of Paris IOTL, the Germans would face great difficulties. What's more, there would be the threat put on the supply lines of the German armies by the lot of fortresses still resisting: with 20,000 men instead of 180,000 men to feed, Metz should resist longer, while Strasbourg would not be demoralized by the disaster of Sedan.