Ah, my apologies for the confusion then. I thought we were all on board with the scenario outlined largely by Eurofed earlier in the thread featuring a war over Rome in 1869.
I assumed we were talking about 1867 due to the Battle of Mentana which Garibaldi wins with covert support from the Italian government. This would lead to the Pope fleeing to France, a huge crisis and then a war with Italy refusing to give up Rome and Prussia supporting them. This and Luxembourg of course, I can't imagine the issue of Luxembourg would've been written off yet as I can't imagine Napoleon III would be so ready to back down over the issue.
As towards the Mitrailleuse there were few of them and their crews poorly trained IOTL due to the massive restrictions the monarchists Senate imposed on the military. ITTL though that doesn't happen, and both Louis-Napoléon and Eugénie were avid modernists and industrialists. So ITTL there's likely to be more Mitrailleuse, and better trained crews operating them; or at least a device similar in function and scope. At least by 1869; see below for 1867 commentary.
I still don't see how the Monarchists won't get as much power here? If your claim is correct and they wished to bring Napoleon III down then surely an even bigger failing in 1866 would simply boost their support.
Well now at least we're not discussing IOTL. I have to ask though why you believe ITTL France would be defeated by the Prussians and Italians in less time than IOTL, considering the differences between IOTL & ITTL.
I'll deal with this below
Asides from the Francophile claim all valid critiques; and now the discussion is moving forward at least.
The Chassepot is a clearly superior weapon to the Prussian's Dreyse needle gun, with something like twice the range iirc. Now, if we're talking a 1867 war (over Luxembourg I assume), then yes the discrepancy is going to be in Prussia's favor due to the limited number of Chassepots fielded by the French. However it wouldn't be a cake-walk even then, as the weapon the Chassepot was replacing was the Tabatière rifle, a breech-loading conversion of the muzzle-loading Minié rifle, which had a much lower rate of fire compared to the Dreyse, or even the Chassepot, but still had a much greater range than the Dreyse.
True, however IOTL the Chassepot, while greatly helpful to the French cause in killing many Prussians it didn't save the war for the French. Now I know this is a different TL however a weapon that's not as good as the Chassepot (to me) can't help them anymore.
In 1866 France could field 250,000 troops within two weeks, including active and reserve units. Louis-Napoléon wanted a million troops at his disposal, and so by the Franco-Prussian War France had over 800,000 soldiers. An 1867 war ITTL means French numbers will still be on the low side, probably around 375,000 troops, taking into account Louis-Napoléon's stronger position and Eugénie's hardliner approach.
Comparably iirc the Prussians, and their German allies, had fielded 500,000 troops in the Austro-Prussian War, but the early battles of that campaign featured Prussian armies in the low thousands until the decisive victory at Sadová, nearly a month after hostilities commenced. If we're discussing a war over Luxembourg this could possibly mean the French take the city and press into the Prussian Rhineland before the Prussians and their German allies are able to bring to bear the full weight of their numbers.
Ah, but now I must tell you to remember this is a TL

The Prussians (i.e. Bismark) would've learnt from this mistake and surely now, as they'd been training for for years, would have a system in place for effective, large-scale mobolisation with the use of railroads. Also on Luxembourg it was described as the 'Gibraltar of the north' - And ITTL it's still occupied by Prussian forces, while I do reckon that initially the French would siege the place when the Prussians quickly get to Luxembourg they'll push the French away.
The Italians numbers I'm not entirely sure of. Wikipedia has the Italians listed as fielding some 120,000 men at Custoza, but then less than 40k at Bezzecca. This is obviously a large difference and could play a pivotal role in the outcome of the war; especially if the Italians are fielding large enough numbers the French have to divide their forces to keep the Italians from Lyon or Marseille while occupying Luxembourg against the Prussians. Even then however the Italian war effort is going to be quite lackluster; Garibaldi was more of a guerrilla warfare specialists than he was a suited for traditional European combat, especially by the late 19th century. In general the Italians in this period were notoriously bad at great power warfare so I think we can agree on a limited southern theater in the Alps and the mountain passes of Savoy and Nice that's going to see the Italians launching rather weak and poorly organized offensives against a strong French defensive position.
According to Wikipedia in the whole war the Italians had 300,000 troops. As Eurofed has been saying the Italian across the Alps would be slow and wouldn't be an immediate thing, however the French would have to spare some of their 375,000 troops for that front, probably a good deal once the Italians get through. Not only that but, as you say, Garibaldi will be involved and is good at guerrilla tactics, don't forget that Nice is still primarily Italian and was Garibaldi's homeland. I can imagine there'd be a lot of guerrilla warfare within Nice, I expect the French would lose as well while having to deal with an approaching Italian army and no-more troops being spared from the Prussian theatre. Not only that but Savoy wasn't too happy about the annexation, though it is far less Italian, I reckon we'll see a minor 'civil war' within Savoy which I expect the Italians will win with help from Garibaldi.
After this we'll probably see Garibaldi go to Corsica while the Italian army deals with the pitched battles in France, Garibaldi will see limited success in Corsica though I reckon he could succeed if he brought a large enough army with him and managed to whip the Corsican's into rebellion if he can convince them France will lose thus the way forward is with Italy.
Overall Napoleon III will lose as he's splitting his 375,000 strong army in order to deal with a Prussian force that could be 500,000 strong
or more which has better artillery, also Napoleon doesn't have the use of a machine-gun or amazing rifles (though the French ones were still a bit better than the Prussian ones). Not only that but the French will have to deal with guerrilla warfare in Nice, Savoy and (possibly) Corsica to the south while an Italian army between 160,000 - 300,000 invades their southern cities and sets up sieges of places like Marseilles. Also the Italians, while not great, would have high morale as they're continuing their successful, and to them inevitable, Risorgimento.
On the issue of veteran troops, both sides will have veterans. The Prusso-Italians fought a war just the previous year and were highly successful in it, while the French have around 40,000 veterans from Mexico, however other than that their other veterans are minor numbers of troops from various expeditions, most of which happened years prior to the POD, at most some 1,600 troops from Korea and the Battles for Shimonoseki. Don't forget though that the troops from the expedition were against rather undeveloped troops while the Mexican veterans were supported by many of the Mexican natives, here it's the French against well-trained, veteran Prusso-Italian troops.