The Treaty of Warsaw left Russia greatly destabilized. The current government was hated by the populace and from 1711 to 1714 there were many uprisings against the Tsar's government.
An implication that Russian population gave a damn about that war (outside understandable wish of it ending ASAP) clearly indicates what you, with a laudable honesty, confessed: unfamiliarity with the issue on which you are fantasizing.
These last three years of Peters reign were some of the most tumultuous in Russian history. Tsar Peter would become incredibly hated in these years
I can tell you a big (and commonly known) secret: in OTL he was hated as well and his reign was as tumultuous as it goes.
and is still viewed as a stain on Russian history with many in history and now referring to him as Peter the Dishonored or Peter the Foul.
And why would that be? His grandfather and father had been losing to the Swedes as well so nothing unusual there. And why bother with these foolish nicknames if a noticeable part of the population clearly considered him being an Antichrist?
The number of Peters supporters quickly fell during these years as people believed Russia still could have won the war
Sorry, are we talking about Russia of the early XVIII? I don't think so. AFAIK, an overwhelming majority of the Russian population did not care for this or any other Peter's wars because they were producing nothing for them besides troubles, death and misery. An idea that there was some kind of a "patriotism" involved is simply bizarre.
and they disapproved of his tactics in subduing the uprisings that plagued the nations.
How original! Did they approve the tactics used by his predecessors in these cases? Or perhaps in your AH Peter was trying to convince the mutineers and forbade usage of the force? Well, this could definitely destroy his reputation on
both sides of an uprising.
By the last six months of his reign only a handful of people in the government and the military supported him including Alexander Menshikov who would lead a small force of Romanov loyalists against the Sheremetev's forces just after the coup in 1714.
Boris Sheremetiev is approximately as likely candidate for being leader of anti-government coup as I'm a ballet dancer.
The beginning of these troubles was when news of the treaty reached Moscow. Almost immediately people took to the streets and a mass riot started.
Yes, of course. The people much preferred to pay for the pointless war (or to be sent to die in it) to finally getting some peace. Of course, they were rioting on the streets demanding to be recruited into the army and to fight until the glorious end. Or to be sent to the swamps of Ingria to build Peter's "paradise".
If you are writing nonsense, at least try to be a little bit more inventive. Something along the lines

p):
..... All population of Moscow, including women and children, armed themselves with the pitchforks and axes and started marching, under the leadership of Martha Posadnitsa and Ivan Susanin (*), westwards with a battle cry "Даешь Европу!!!". On the route they had been joined by the population of the areas through which they had been passing and kept growing in numbers. By the time they reached Smolensk (which seems to be the only Russian city, besides Moscow, that everybody knows about), their numbers rose to couple hundred thousands. The Poles, who had been happily celebrating return of the city, did not have time to sober up .... and the crowd kept marching all the way to the Baltic coast. Peter and "a small force of Romanov loyalists" tried to join the movement but had been rejected due to the improper attire. It took them couple weeks to grow the beards, get dressed into armyaks, porty and onuchi and then to catch up with the crowd. When they almost reached the border the marching crowd had been joined by a bunch of the Russian alt-historians from the XXI century itching to change the past and armed with all hand-held weaponry they could procure. (**)
Charles, who was still hanging in Livonia trying to figure out how to entertain himself after the peace was signed, jumped to opportunity and marched to meet the invading horde. Unfortunately, the Swedish bravery was quite useless against the rocket launchers and AK-47s... The crowd kept marching westward. Charles's severed head on a pike served as their battle standard....
By the time they reached the Baltic coast, it was already winter. With Daugava frozen Riga was an easy victim and, after fortifying themselves with its Black Balsam, Rupjmaize and Sklandrausis, the crowd, now proudly calling themselves "AHLA" ("Alt-history liberation Army") kept marching north to the frozen Gulf of Finland which they crossed by ice to invade Finland. Within days all local supply of "Finlandia" had been exhausted and AHLA was ready to march across the frozen sea right to Stockholm where they expected to find an unlimited supply of "Absolut". Seduced by the perspective of getting a free Swedish drink quite a few Finns joined AHLA. However, even before they reached the Aland Islands, a huge sledge train from Sweden carrying barrels of Absolut and a humble request of peace on any conditions met them.
The ensuing Aland Peace was harsh but better than the alternative of being left completely without a booze. Russia got Livonia and Sweden promised to keep selling "Finlandia" and "Absolut" to Russia at a discount price. Being upset with what they considered to be Swedish treachery ("these Swedes are selling our national resources cheap!") and seduced by a promise of cheap "Stolichnaya" and 100 barrels of free pickles, Finland rebelled and united with Russia.
To avert seemingly unavoidable revenge Sejm of the PLC (***) ordered to send all existing supplies of
Gdańska wódka, Zubrowka, Luxsusowa and Sobieski to the Russian-Polish border as a token of the good feelings.... The resulting feast was a foundation of the ever-lasting Russian-Polish friendship.
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(*) These two would do just fine for the people unfamiliar with the Russian history.
(**) I found that there are numerous Russian fans of the genre who are considering "rewriting history" as a part of their patriotic duty.
(***) According to the national experts, the PLC was too busy celebrating the death of King August (who fell from a balcony and broke his most important body part) to elect a new king.
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