Liechtenstein would likely use Alaska as a port to the pacific, especially with the likes of Russia, Japan, China, etc. Multiple Liechtensteiners would likely move to Alaska hoping to hop on board with the wealth. During WW2, Liechtenstein would probably be seized by Nazi Germany, so the Liechtenstein monarchy would use Alaska as a National Redoubt. Japan would attempt to take the Aleutian islands to some success. After WW2, Alaska would gain independence, with Liechtenstein keeping the Aleutian islands. (Similar to the British Indian Ocean Territories.)

I'm working on a timeline where the House of Liechtenstein gets the Greek throne during the 1832 London Conference and accepts the Russian offer for Alaska, thus sparking a colonial empire that would encompass OTL Italian Somaliland, Socrota, Attu Island, Alaska, Libya, Syria, Antioch, Turkey, Kurdistan, Greater Armenia, Crimea, and Azerbijian. I'd be very interested in working with you!