I did make quite some search on the topic but for some reason this story escaped me. Thanks for illuminating me on this anecdote.
On the subject of a sale to Liechtenstein, I doubt the reasons for the failure of OTL would magically disappear. Alaska was still a "frozen wilderness" at the time with not much value, and it was prohibitively expensive to hold onto and administer. Even though very rich, it's rather easy to understand why Liechtenstein ruling family wished to remain so. In my sense, it would take someone craving for oversea expansion and near desperate to pursue this goal. In a way, Seaward was, but if you want someone in Europe, I'd suggest Leopold II of Belgium, given the lengths at which he went to acquire a colony, getting and losing Congo in the process.
On the form though, assuming the deal unexpectedly goes through, then I doubt Liechtenstein would hold the territory as far as WWI, even if a de facto Austrian colony. I think it's more likely it would be taken over by Anglo-American settlers, filibustered like Hawaii was, prior to annexation by the US. Gold is only going to excite American appetites, and unlike Russia, Liechtenstein hasn't the shadow of a chance to hold onto the province, and Austria-Hungary has neither the proper claim to push against an American takeover nor the material means for a military adventure to reclaim the area, and no European continental power is likely to go to war over this issue, especially after Spain got utterly crushed in Cuba and in the Philippines. In the end, that wouldn't make Alaska much different in shape than what is today, other than stuffing history books about the state with a bit more curiosities and color.