This is a future TL, so the POD is now. Soon after the purchase, the US and Canadian governments built a bridge (well, three) across the Bering Strait, and roads to connect it to Magadan and Fairbanks, so it is possible to drive from (say) Paris to New York (although it'd be easier fly or take the hyperloop).Why would Canadá which till 1931 was a subordinate nation of Britain which had no true foreign policy buy Siberia? Sorry it would not have the $ or ability to defend it. For over 20 years after building of the Canadian pacific railway the country had serious cash flow problem and a deficit. So I finding that historically not feasible. That Britain bought it and transfered to Canadá more probable but again not very realistic.
this is your idea I just providing historical context. Also why US buy Alaska and not Britain or Canada if Russia going to sell they be more inclined to sell it than Siberia. Plus with stronger British Columbia the Americans be weaker in pacific.