I'm dubious, but it's a great story if you don't look closely.

The article and author you provide are sketchy in a different manner - he makes some basic mistakes on how the purchase actually went down that even tertiary sources don't - but he does include a link to a newspaper where it turns out the source is the current Prince of Liechtenstein, Hans-Adam II, who went so far as to send a letter to some of the local papers to confirm that he's made the claim.

It's in German and requires registration, but a summary is this: his ancestor Prince Franz, an Ambassador to Russia for Austria-Hungary and later ruling Prince of Liechtenstein, was close with the Tsar, who needed money to expand into Siberia. The family had the money and the Tsar was interested in selling, but then decided against it since the distance made it impractical. It is apparently a story that's been told down the years by members of the family, usually with regrets about the gold found in Alaska. There are no written records; the Prince argues that it was either early stage and/or informal enough so that it never made it to paper, as well as that some of the family archive was confiscated and/or destroyed by the Soviets at the end of World War II, with what portions that have been returned having not been fully reprocessed.

Pretty interesting, and the Prince reiterated part of this story again in a later, English language interview with the Economist.

Except the first part has one basic problem: the math doesn't add up. Prince Franz was born in 1853, which would have made him all of 14 when the deal with the United States was consummated in 1867, and presumably even younger when this theoretical discussion would have taken place. This does not make it completely impossible - John Quincy Adams was the same age when he first went to the Russian court in 1781 as an interpreter, although he was viewed (properly) as a child prodigy - but a complex negotiation seems a bit of a stretch, and if you look at the Times article on his appointment as Russian Ambassador in 1894, Prince Franz's early years were spent at the Legation in Brussels, not St. Petersburg.

The second part is based on something I've written previously on the details of the Alaska purchase, part of which I'll repost here:

Meanwhile, (because of impeachment proceedings against Andrew Johnson), everything else on the Hill ground to a halt for months (on top of being a year after the deal was concluded.) This presented a dilemma for (Russian Ambassador to the US) Stoeckel, who actually wrote back to Russia after an inquiry that they could either wait (his preference) or just outright offer Alaska for free! In May, the House Foreign Relations committee - with a number of supporters of the Perkins claims who noisily dissented - recommended approval of a payment, but that debate got stalled until July and the final 113-43 vote didn't take place until July 28th.

It seems very, very unlikely that if there was any sort of potential second bidder for Alaska that Russia would have tolerated all the twists and turns of the negotiations and the Congressional mess that had to get cleared up before they got paid anything for it, at least without ever trying to provide a sense of urgency to it by hinting that a second bidder existed - let alone being presented with a scenario by their Ambassador in which they would have potentially given it away for free.

So no, I would suspect this is a family story that's been handed down for 150 years that, like many yarns, has gotten a little exaggerated in the telling.