From my reading of Castlereagh's foreign policy, I don't think the Holy Alliance was anything like what Castlereagh wanted. Was it a group of European states? Sure, but it was also an alliance that wanted to impose its will, nothing like the Congress system that allowed equitable agreements about foreign policy problems to be reached. By the time that Castlereagh killed himself the Congress system had already fallen apart- an event that probably was a major cause of his pen-knife incident.
Castlereagh on his own is still going to be a force to be reckoned with in European foreign policy, because of his long association with the Congress system, and thus the knowledge of other European powers that he is an honest broker. Whether he will be in government is another question. His foreign policy was unpopular, his government unpopular, and he was personally unpopular.
So I would say Castlereagh is going to have to retire from his role in government. He retires in 1822, citing his declining health. George Canning succeeds him as both Leader of the House of Commons and as Foreign Secretary.
Castlereagh's retirement lasts from 1822 until 1827, when the Lord Liverpool dies. George IV, who repeatedly and publicly opposed Catholic Emancipation, gives the Preimiership to the Castlereagh. Canning refuses to serve under him, but Wellington, Peal and other 'Ultras' support Castlereagh.
The fall of Castlereagh's government was a fraught affair. Castlereagh's Tory government refuses to accept Catholic Emancipation, and instead tries to suppress unrest in Ireland with force. Growing calls for further reform in Parliament however only strengthens the resolve of the Ultras. Elections following the death of George IV produce larger than OTL Whig gains.
Castlereagh, his health again declining, decides that a deal needs to be made, and makes a deal with Canning- Canning will become PM, and Castlereagh his Foreign Secretary. Wellington and Peel, following Castlereagh's lead and despite their opposition to parts of Canning's agenda, decide he is a better alternative than Earl Grey, and support Canning's government. Canning, with Castlereagh's and William IV's support is able to produce a Reform Act that is acceptable to enough Tories to pass it.