The French candidate did not much want to take the throne, and Augustus III was a small child at the time. With the rest of Europe busy with the Spanish succession, I think Jakub Sobieski would be the most likely candidate, particularly if he promises to desist from the war against Sweden.
If Sobieski goes even further and aligns with Charles against Peter (perhaps implausible, given the Sejm’s laziness, but Charles’ charisma and victories could sell them on it), the two have a good shot at beating Moscow entirely. If Peter can be killed or captured, and his son Alexei put on the throne, they could put Russia back where it was in 1610—close the ports and shipyards, put the most regressive people possible in charge in Moscow, and carve Ukraine and Smolensk back off. The Swedes had a great record against the Muscovite army for the first chunk of the war IOTL—more pressure could break them.
Sorry, which "ports and shipyards" are you talking about? Within that time frame Russia has 3 ports: Astrakhan on the Caspian Sea, Azov on the Sea of Azov, and Archangelsk on the White Sea. The only one within a reach for the Swedes is Archangelsk but the Swedish attempt to attack it in OTL (1701) ended with a loss of 2 ships (out of 10) and retreat. Taking into an account that this port was important for the Brits and Dutch, I would not bet too much on the repeated Swedish attacks.
Then goes the "1610" thingy. If anything, Sweden was
not interested in a "total war" with the Tsardom because it was greatly benefiting from the existing trade agreements: Swedish-held ports on the Baltic Sea had been outlets for the Russian exports and Sweden was getting the custom dues (even with the allowance for Charles' idiocy, he should be aware of the basic facts of life). Charles wanted restoration of the pre-war status quo, not any serious redrawing of the map. Look at the OTL examples: this was his pattern with Denmark and August (removing him from the throne of the PLC was a logical way to kill the future designs regarding Swedish Livonia but Electorate of Saxony was legally his and Charles did not even try to change a dynasty).
The PLC was Swedish traditional enemy (it is easy to check how many wars they fought in the XVII century) and, adding to this fact, Polish Catholicism, Charles' personal negative attitude and almost absolute military uselessness as an ally (in OTL by the 1708 Charles had his puppet king of the PLC but I don't recall any Polish military involvement in Charles' Russian campaign of 1708/9), it is rather naive that he would do anything on its behalf (sorry guys, I appreciate your patriotic feelings but let's be a little bit realistic

).
By 1700 the high-placed "regressive people" in Russia could be found mostly in the movies and books of Stalinist period (should I comment any further?): top nobility was quick to understand advantages of the Western life style (in "Muscovite interpretation") which was gradually introduced since at least the reign of Feodor II. His main political opponents, Princess Sophia and Vasily Golitsyn, were still alive (even if safely away) but here goes the problem: both of them had been quite "progressive".
Chance for Charles to impact Russian internal affairs was approximately as high as his chance to conquer the Tsardom: close to zero. Sweden simply did not have the needed resources and the PLC of the early XVIII already was a military and political joke.
Ukrainian part belongs to the same category: Charles did not have a slightest intention to give it to the PLC when he was on Ukrainian territory and, while it is possible to speculate about the future of the Hetmanate (in OTL it took just a sack of Baturyn to kill the whole idea of independence), getting back under the Polish rule already was not in the cards: the Cossacks would resist.
Charles would be most definitely beating Russian troops in the field for the next few years (as he was doing in OTL) but the problem remains: his military force is too small and the distances are too big. The "peripheral" Swedish detachments would be beaten as they were beaten in OTL in Livonia, at Kalisz and at Lesnaya and the new troops raised in Sweden would not have quality of Charles' veterans. He could not be simultaneously everywhere which means that at least some of the important places in Livonia are being lost.
His Russian expedition was an act of a desperation (and would be even if it happened few years earlier). His army at that expedition was his biggest army ever and he was quite successful on the initial stage of campaign by defeating Repnin at Golovchin. What happened? Pretty much nothing (except that Repnin was temporarily demoted): the Russian army kept retreating burning everything on its wake and forcing Charles to march on Ukraine where he expected to get some supplies (making Lewenhaupt vulnerable to the Russian attack).
It is not that Peter was such a good general (Sheremetiev was not a military genius either and Menshikov was ...er.... "energetic") but it would not take a genius to figure out the correct course of actions. Charles was a very good tactician (with a somewhat overblown contemporary reputation of being the best, which IMO was not the case) but absolutely pathetic strategist and witless monarch (stubborn as a mule). He was doomed to fail against the opponent who had much greater resources, a lot of space for maneuver and no place in his state about which he really cared (see comments made by Fieldmarshal Montgomery on this subject).
But that assumes Peter does not make peace. This is before the foundation of St. Petersburg, so at this point, he has little to actually lose by agreeing to a cease-fire. If Augustus is dead and Denmark beaten, it’s equally likely for Peter to give up, aborting the war entirely.
This is a very good point which I keep repeating without visible effect.
Being somewhat obsessed with a notion of "justice" Charles was looking for safety of the Swedish territory, not for the conquests. In 1701 Peter's gains in Livonia are still minimal and he can agree to the "peace without annexations" with a relative ease. IIRC, in 1708/9 he was ready to give back ALL conquests in Livonia (which were really valiable) on condition that Charles will leave him Ingria (with the site of St-Petersburg). Of course, Charles rejected this proposal. In this AH Peter does not have to give away anything he really values, just a looted countryside of Estonia.