Okay:
1. Russian Policy
Russia is entirely focused on avoiding another war with Britain. The Crimean War was an absolute disaster for Russia, who lost more dead than both sides in the entire ACW. In 1863, the deployment of a couple of small squadrons overseas is insurance against the Anglo-French alliance deciding to intervene in Poland.
2. Russian Navy
It's small, very small. They'd lost a vast portion of their fleet in 1854-6, and still hadn't recovered. The force sent to New York would be roughly handled by the Coast Guard Squadron off Queenstown (Cork), let alone a British cruising squadron.
Also, the British had forced the Russians to dismantle large parts of their coastal defences in 1856. Any local RN squadron could dominate the coasts.
3. British Intervention
The British came close to intervening in the ACW 5 times (the last time in July 1863), it doesn't require a lot of handwaving. Maybe the San Jacinto is intercepted by the RN at the time of the Trent Affair (British intelligence had penetrated the US to such a great degree that Milne had spies actually aboard all major USN vessels, he knew of San Jacinto's intensions (even though the US government didn't), and had ordered any RN warship encountering her to shadow her and engage her if she approached any British Packet)