Possibly, Glen, possibly, but...I think not. The British were the merchants of the world at this point. Not only did everyone expect to see them everywhere, and not only were they often the best way of turning a profit from a bad situation by their capability of taking any goods and finding somewhere in the world where it was needed, for the highest profit margins, but more importantly was the fact that they had the reputation for being able to do the above. Even in cases where they couldn't sell a product and were ripping off the original traders to guarantee that the middle man made money, the original seller felt like they were onto something, because no other nation could guarantee trade with the rest of the world. The Americans, by contrast, had large parts of the world where their merchants simply did not go. Not only this, but this is Russia we are talking about. The same Russia which was highly bigoted in its dealings with Europe - they, for instance, believed that they had the sole right to represent Slavic peoples, above and beyond the existence of other Slavic nations, and they started their own war in Europe when they tried repeatedly to take Constantinople, claiming that it was their right to govern such an important eastern city. I suspect that, if the Americans turned up and asked for a Most Favoured Nation agreement with Alaska, the Russians would laugh in their faces and say "who is America?"
[ref to Europeans jumping in on the act over Alaskan saber-rattling]
What?
Over Alaska? Maybe...but somehow I think not.
OK, OK, I portrayed that badly, but I stick to my theory. The point is, it's different when the US is trying to attack European colonies. In OTL they didn't try it until the 1898 Spanish-American War, by which time they had had opportunities and the far side of 120 years to prove who they were and to make a name for themselves as a true growing power. In TTL, they are yet to do anything ground-shaking to prove their might, and yet we are advocating that the first war they intend to try is against a European power which had been ignoring them, and who they shouldn't have had a problem with. To much of Europe, this is going to smack of a jumped-up ex-colonial freshman at school (to use the American slang) being told by one of their nerdy, stupid, foreign-exchange student friends that the only way they can win respect is to beat someone up, and rather than picking on a fight with another freshman who no-one cares about, he picks a fight with Russia's weak brother, who isn't able to defend hisself. Not only is this an insult and a direct attack on Russia, but it's a statement to all the European "jocks" that America, if it gets away with it, intends to keep building a reputation "or, to go back to our actual scenario, "cleaning up those borders") by beating up the rest of their kid brothers too. After all, if Europe is going to stand by and laugh while Alaska gets picked off, why on earth should Europe feel surprised if America then feels empowered into attacking, say, French Guyana, or the Philippines, or Portuguese Africa, or even the DSA.
At least in OTL, when the US attacked Mexico the European countries could look at themselves and with reasonable confidence say "yeah, but they would never dare attack us, so our colony is safe". Yet in this scenario, the USA's very first expansionist attack is on the European elite themselves. I'm not suggesting here that the other European states would join the war, but I do feel they would respond by sending the USA messages to the tune of "what do you think you're doing? Do you realise how much we can hurt you if you try this on us too?"
Finally, and this is an important point here too, which I dreadfully undersold in my posts, the means of the American attack in a situation where Russian Alaska officially has closed borders is important. It is different (not much, but it is) to send over colonists over an open border, have them legitimately take up possession of the land, and then when they are mistreated call for their home government to support them, and the result of a war being that the US legitimately demands that territory as compensation. In this scenario, US settlers were never allowed to go to Alaska. Yes, they might be able to sneak there and start up some mines (let's not forget here, though, that Alaska is woefully under-populated and is very rarely visited by ships aside from British and Russian merchants and the Russian Navy), but even if they do they will be illegal immigrants. It's one thing demanding territory because a load of settlers have moved across the border and feel that they aren't being respected: that's gunboat diplomacy, although even this is the extreme end. But in the Alaskan case, the US would be demanding the ceding of territory not for this reason, but for the reason of "Russia wouldn't allow us to illegally plant illegal immigrants for the illegal reason of illegally mining and exploiting Russian gold deposits!" I know that a lot of reasons for war and conquest in this era were patchy and questionable at best, but there really were limits to how far you could push it. Claiming that you demand Alaska as compensation for your settlers being treated roughly because they were attempting to break the law in the first place is just going too far, and it would be impossible for the US to convince Europe that it was anything other than "nice territory, I'm taking it"... If you let the US get away with it once, then you should expect the US to start declaring that all European colonies are up for grabs, and it's this reason that France, Spain and the UK would consider threatening the US with sanctions. Remember, this US wasn't a powerhouse in this era, yet, so there's no reason that Europeans - especially France and Spain who have little reason to pay attention to the USA at this time - would treat the USA at this point as a serious nation worthy of enough respect to not warn them over such blatantly expansionist methods.
