1. Beckwith is not a climate scientist, his guesses are as good as mine at the minimum. Actually, he is worse, as he's been known to make extraordinary claims that proved to be completely false. I would be very careful about anything he says.
2. The research he quotes above is valid though, written by actual climate scientists. I read the paper.
3. Ice albedo feedback is a very important but not very well quantified mechanism, there are still a lot of questions about it (mostly cloud-effects). Still, it is unquestionable that the warming of the planet (especially the oceans) leads to less arctic ice, which leads to more heating of the arctic sea, that leads to less arctic ice. Eventually a BOE will happen and it will seriously effect the climate of the whole planet.
4. I would argue that as the Arctic becomes icefree, there will be more moisture and more clouds, that lessen this effect, but as all real climate scientists know, future cloud-effects are the greatest uncertainity in modelling climate.
5. Most of the extra "heating" would come from spring and summer insolation. An ice-free Arctic in September or even August would not change much as per the paper.
6. Their calculation of 25 years of extra Co2 would probably imply 0,5-0,9 C of global warming. If there are more clouds as I expect, it is probably around or below 0,5 C. All this will happen only with an ice-free arctic during summer and spring! That is basically a year-around free arctic. No real scientists see that happening within 50 years.
7. So for the next decade the extra warming even from an Arctic that becomes mostly ice-free in August/September (this will probably happen in the 2020s I think), is at most +0,1 C globally.
8: It is quite obvious though that +1,5 C or +2 C globally is not achievable. +3 C is the best that mankind can hope for by 2100. I think that is achievable. Mind you, that would probably imply +6 C on NH midlatitude land! (cca +4 C vs current temps)