What if Kościuszko won the War in Defence of the Constitution?

At a guess, the Second Partition would have happened on schedule or even early, with an active Prussian intervention to support Russia in return for a piece of the spoils. Historically, Prussia rather wanted as much of Poland as they could digest. Russia's original intention in the intervention was only to prevent the reformists from succeeding in their ambitions; it was Prussia, combined with the dictates of Napoleonic power politics, that requested and was rewarded with the partition, their price for abandoning the Polish-Prussian Pact of 1790 and a reward for their support to the War of the First Coalition against France. If the Russians look like they'll lose the war and, with it, Poland from 1792, they'll be more willing, not less, to sign away part of Poland to guarantee control over the remainder. Moreover, historically, when Kościuszko staged his little uprising in response to the 1794 partition, this is exactly what Prussia actually did - they invaded right alongside Russia, and his failure there led Prussia and Russia to decide that an independent Poland was more trouble than it was worth. If Poniatowski and Kościuszko drive off Russia in 1792, it may well be that the Third Partition never happens, because the Prussian intervention and the resultant Second Partition spells the end of the Polish state. It might be that the biggest side-effect is an enlarged Prussia at the expense of historical Russian gains (and even that's no guarantee), but with the Congress of Vienna coming up (it's unlikely to have a profound effect on the Napoleonic Wars), that's not likely to be lasting too long, either; Frederick William will be likely to sign away most of these gains to Russia in return for equivalent gains in Germany, just as he did historically, still leading to the same Congressional Poland.

As for how he could have won...that's a toughie. We didn't really get to see what the Polish army could manage in 1792 because the King accepted peace before they gave battle. The uprising in 1794 demonstrates the real issue, though, that even with the mass support of the peasantry, the Polish rebels could never actually take the fight to the enemy. You basically see this happen in that uprising over and over again, where they keep winning battles, but ultimately can't stop the Russians and Prussians from raising new army forces, so their "glorious victories" keep getting closer and closer to Warsaw. If he had succeeded at Maciejowice in beating one of the two major Russian armies in isolation, he still would have had to deal with the other Russian Army as well as the Prussians, which by this point was trying to defeat the uprising in Greater Poland. In 1792, in a straightforward war in the name of King and Crown, his more Jacobin sentiments are likely to be undermined or restricted (after all, there's nothing stopping the King from switching sides, as historically happened), and he won't be able to call on the same patriotic fervor of the peasantry as a consequence.

Hmmm? I think I see the mistake. There were two conflicts, not one, for all that they're closely related. In 1792, there was the Russo-Polish war, in which the Russians invaded in support of the Targowica Confederacy, and in which the king's nephew Józef Poniatowski led the army with Kościuszko, among others. In this war, the King signed a peace treaty, and the war came to an abrupt halt (though there was talk of Józef taking the King, bringing him to the camp under "protection," and continuing the war). Kościuszko was never captured in this campaign, and served only as a military leader. Soon after this, the Second Partition (1793) reduced Poland further, in direct violation of Russian promises. In 1794, the Russians and Prussians together decided to demobilize the Polish military and integrate it into their own, for their own protection. This led directly to the Kościuszko Uprising of 1794, in which he took a lead role, was captured, and was sent to St. Petersburg. The uprising, however, continued even after that; it's just that his successors (Wawrzecki militarily and Zajączek politically; the king himself was effectively irrelevant at this point) weren't up to the task of keeping the fractious rebel fractions together politically, and in either case, were compelled to surrender in November after military defeats. Two weeks later, the King was forced by the Russians to leave Warsaw for St. Petersburg himself, and within the year, Poland ceased to exist. There was no peace, because as far as Russian and Prussia were concerned, it wasn't a war to begin with; it was just putting down some rebs.

If Kościuszko hadn't been captured in the 1794 uprising, he would have been free to prosecute the war with the king's unlimited support (insofar as that meant anything), because it's very unlikely the king would have signed a second treaty after what happened the year before. If he and Józef Poniatowski had kidnapped the king and continued the war in 1792, it's more likely that they would have done it in the king's name (protecting him from the vipers of the Targowica Confederacy who forced him into it under duress, or some line like that), or else made Józef himself the new king due to his blood ties and the fact that the present king had no legal issue, rather than declaring a republic. In either of those two cases, with complete control over the Crown and powers of government, they probably would have liberated the peasantry and turned it into a national war of liberation as per both of their wishes, at which point it would probably just go right into a replay of the Kościuszko Uprising and its ultimate outcome. Even if they and the Russians want to keep the Prussians out, it's unlikely the Poles in Greater Poland would desire that, and an uprising there would make it a Prussian interest as well (not that they even need an excuse). The only way they can survive is to keep winning long enough for Napoleon to become a bigger threat than the Polish side-show, and even that will only last so long as Napoleon remains a threat. For the same reasons they historically did, it's likely Poland would still side with Napoleon, and if he loses, Poland still loses its existence. If they stay neutral or side with the Coalitions (say, after Russia and Prussia are compelled to side with Napoleon, but before Napoleon invades Russia), then they probably still get eaten in the end; if it's not as the price for Prussia and Russia to support Napoleon at Tilsit, then for the price for those two powers to accept peace at the Congress of Vienna during the inevitable Poland-Saxony crisis or its equivalent. Remember, the historical resolution of the Congress of Vienna called for an "independent" Poland with the Russian Tsar as sovereign, and we all know how long that unstable situation lasted. Sadly, I think 1792 was simply too little, too late; it would have taken a miracle, and not a small one, to preserve an independent Poland against both Russian and Prussian ambitions.