Russia wasn't comparable to Mexico, and I believe 19th century american governments would not dare attack Alaska, for the same reasons they gave up on annexing Canada. It took them until the turn of the century to make a move to take Cuba, which was part of a much weaker (and more diplomatically isolated) empire. They would occupy in during the russian civil war, and probably keep it too. Which would in turn lead to much worst relations by the time of WW2 and after. The russians didn't forgot to retake the territory lost to Japan, would they forget about an Alaska lost in a war?
So we might have been victims of a hot war over a territory at the end of the world!