Guys? Gals? Crazy idea: Russians now know that Alaska has fantastic gold reserves...it also has Britains biggest and most developed colony on its eastern border and is also even less developed than the Russian Far East (which in this time was considered a Land of Yurts by anyone of note) and even with the gold would be a money sink because the mining would necessitate building up defenses and infrastructure for Alaska.
This is the Era of The Great Game. Britain will try something with Alaska. It is not a matter of If but When.
So? Sell Alaska to California while keeping 50-60 percent shares on any future mined gold (Russian leadership should be now aware from history books about late 20th-century globalization that California would always follow a trade deal it signs), with California delivering the mining and refining technology...and paying for the development of Alaska.
AKA: We get the gold while sitting on our asses and you have to deal with the Brits and with paying for the development of the land!
The Russians know that with Californian technology, even though Alaska is not as super gold-rich as some other places around the world (specifically, places inside the British colonial empire), to which they will never get free access to anyway, Alaska will still become the biggest source of gold on the planet for at least two decades to come.
Plus! With California in Alaska, the British will think twice about supporting some kind of insurrection or similar to dislodge the Russians/Californians from such gold motherlode.