A point of diversion that might have more promise is if Russia keeps Northern Iran after Peter the Great conquered it in 1723. Historically, these territories were returned in 1732 as part of a rapprochement with Persia in the face of Ottoman aggression against the other two powers. The Russian government under Anna Ioannovna was also concerned with the cost of maintaining a large body of soldiers in the region. The soldiers often succumbed to diseases endemic to the southern coast of the Caspian Sea, which meant replacements had to be constantly dispatched to keep the garrisons strong. A scheme to build a silk-manufacturing industry also floundered, reducing the value of the region to St. Petersburg.
If you change a few things around to make the provinces more viable from an economic and military standpoint, it could be used as a base to conquer far more of Persia than the Russians did in OTL. Russia's war with Persia in 1796 might turn out to have greater significance then it did historically in such a scenario.
As for Paul's attempt to conquer India in 1800, the less said the better. It seriously undermined the Russian elite's confidence in their emperor, and may have contributed to support for Paul's assassination in 1801. Those Cossacks that composed the expeditionary army were very lucky to have made it back alive, considering Russia had no reliable maps of Northern India or the surrounding region. The army was apparently expected to march into Central Asia and then stumble south until they entered India. Assuming they did not die of starvation in the middle of the desert or the Hindu Kush mountains, I seriously question that the ten thousand-strong army would have the resources to conquer the independent kingdoms of the Punjab, let alone the entire subcontinent. It is doubtful the agents of the East Indian Company would have even caught a glimpse of the Cossacks, let alone have to bring in troops to crush them.