Britain allying itself with the ones who are actively trying to destroy any balance of power doesn't seem very plausible.
With hindsight, yes. Besides that, Britain was the one pushing Prussia past the Rhine onto a border with France. It also makes sense to cooperate with Prussia in regards to Hanover. Mutual understanding can lead a long way. Russia was indeed a wild card, but Alexander's plans weren't well know at the moment & it's more than likely any Belgian issue would be resolved prior to the Polish-Saxon crisis.
Russia with its Polish ambitions and Prussia with its German ambitions don't seem to be the kind to contest Austria and its North Italy ambitions, and Britain is likely still more inclined to ally itself with Austria than with Prussia and, especially, Russia.
Your perspective is too narrow. Just because Russia nor Prussia have any stakes in Italy doesn't mean they would or wouldn't contest it. If it's a means to weaken a rival then it makes total sense to object to their ambitions.
Britain's options are ally itself with the Austrians against Russian expansionism and accept the Austrian Netherlands or ally itself with Russia and Prussia and take down the Austrians who are key in Britain's desire for a stable Continental Europe. IMO GB is more likely to still ally itself with Austria.
You have to realize that Castlereagh's instructions were to not get involved in another European war at all cost. At the end of the day, they're going to put their interests first which was containment of future French aggression--not Russian. This also does not paint the full picture. Britain aligning,
at that moment, with Prussia & Russia doesn't mean they're going to completely throw Austria under the bus at any turn. The feared Russian expansion that you say Britain would not align with was actually appeased iotl, which ultimately led to Prussian designs for Saxony down the drain because Alexander got mostly what he wanted: a kingdom of Poland under a PU with the Russian Emperor, notwithstanding a few exceptions (such as losing the fortresses of Krakow and Thorn). I'm sure if Prussia were the one, and not Russia, to take steps to finding an agreement first, then "Congress Poland" would probably not be a thing and more of Saxony would've been Prussian.