Maybe look at the Democratic Republic of Georgia, where there was a Menshevik government until conquered by the Bolsheviks...
It's not the most representative of what would be like with Russia, but it may provide some insights...
Take from it what you will but Trotsky described Georgia as a 'Menshevik Fortress'. The Mensheviks in Georgia firstly used Cossack detachments and their own soldiers to disarm the soldiers who were loyal to the Bolsheviks. The Menshevik Djugeli described the events in 1918, "This was not a disarming, but a plundering of the soldiers. These unfortunate men, weary, longing to get back to their homes, were deprived of everything, even of their boots. At the same time quite a trade was carried on. The arms were sold to robber bands. What took place was disgusting". These same Cossack battalions were used to suppress the peasants throughout Georgia and the region. Djugeli wrote "Ossetian villages are burning all round us ... In the interests of the struggling working class, in the interests of the future socialism, we will be cruel. Yes, we will. I can look on with imperturbed soul and clear conscience at the fire and smoke of the burning houses ... I am quite calm, quite calm indeed." And the Abkhazian Mensheviks sent a report to the Georgian government describing the detachment sent to suppress the peasants: "This detachment, by its cruelty and inhumanity, has surpassed the infamous Tsarist General Alikhanov. Thus, for instance, the Cossacks of this regiment broke into peaceful Abkhasian villages, carrying off anything that was of any value and violating the women. Another part of this detachment, under the personal supervision of Citizen Tuldiareli, indulged in bombing the houses of those persons who were pointed out by informers. Analogous deeds of violence were perpetrated in the Gudaut district."
The Bolsheviks themselves were suppressed, their meetings disrupted by gunfire and arrests and their newspapers declared illegal, and this all happened
before the reverse had happened throughout Bolshevik Russia. The Mensheviks in Georgia worked closely with the foreign powers. German troops were invited into Georgia to help suppress the Bolsheviks under the command of General von Kress and then, once the Germans had left, British troops under the command of General Walker assisted in the suppression of strikes and peasant uprisings. Walker dictated some of the internal policy of the Mensheviks and an executive order came from his hand: "All Bolsheviks entering Georgia must be imprisoned only in the Mskhet [the jail of Tbilisi], and put under a strong guard." Never mind the hundreds of revolutionaries executed without trial who never got the 'blessing' of seeing the inside of a jail cell.
The Georgians also, despite declaring official neutrality, provided arms and supplies to Denikin and Wrangal and even raised troop battalions who joined Wrangal's forces. The Socialist-Revolutionary Chaikin, who organised the peasants in rebellion against Denikin throughout the Crimea, wrote in 1920 "It is self-evident that such facts as General Erdeli’s free departure from Georgia, the arrival from the Crimea of Denikin’s recruiting generals, who were not interned on their arrival in Georgia, and finally the propaganda and recruiting campaign in Poti of General Nevadovski, and others, most certainly constituted an infringement of Georgian neutrality in favour of the Volunteer Army [Denikin’s forces], and was a hostile act towards those forces which were in a state of war with the Volunteer Army." Nearly 30,000 troops were mobilised from Georgia to join Wrangal's forces in the Crimea, often with the assistance of British or French ships, and the Mensheviks assisted the White forces who, at the time, were massacring workers in Rostov, Novocherkask, Ekaterinodar and elsewhere throughout the region.
I actually think that a Menshevik Russia could have been more stable than the Russia of the Bolsheviks but only on the basis that the Imperialist powers were willing to support the Mensheviks instead of actively oppose them and, just as the Social Democrats in Germany were massacring the likes of Rosa Luxemburg, only over the bodies of thousands of workers and peasants.
