Only in Anglophile dreams.
After 1815, Austria had managed to get a decent deal for themselves, and had been decisive in pushing Napoleon away.
Austria should be more worried about France, and what it means for their Italian possessions.
A large section of this forum consists of Anglophile dreams.
I agree that a France making aggressive moves towards northern Italy will alarm the Austrians and against the Rhineland would alarm the Austrians and Prussians, in which case British diplomacy would have a lot to work with. However per the OP the Second Empire is being quiet and conciliatory in continental affairs, directing its efforts towards overseas/colonial affairs. Therefore Austria, Prussia, and Russia have absolutely no reason to get involved; their interests are not threatened. So in this scenario Britain is deprived of one of its main anti-French weapons, namely paying someone else to do the heavy lifting for them.