I'm thinking it would remain Russian, because if the Americans don't want it and the Brits can't be bothered, no one else will part with even a small amount of cash for the place.
In 1867 Alaska was worthless, remote, and inhospitable. The Russians colonized it in order to harvest furs, and they did such a good job the fur bearing marine mammals they depended upon were driven to near extinction. It was indefensible for anyone other than the British or the Americans, and it was too remote for other powers to even harvest the fish there.
A similar situation existed with Florida prior to the US annexing it by force under Gen. Andrew Jackson. In the opening of the 19th century, Florida was worthless, remote, inhospitable, and ruled by the Spanish who could not be bothered to actually govern it. Florida became a haven for pirates, bandits, escaped slaves, and other outcasts and undesirables of the SE US and the Caribbean colonies of the European powers, and cleaning it up is why Gen. Jackson invaded (against his instructions from Washington DC --- the US government did not want the place).
So, I would think Alaska would become a neglected Russian colony, governing itself because no one else is interested in the place, being populated by Alaskan Natives, Inuits, a few residual Russians, and a motley crew of outlaws and pirates fleeing the law in their home countries.