Breach of the Peace is a common law power that anyone can exercise (theoretically) and it is not a criminal offense. It's not a catch all way to get rid of pesky protesters no matter what that annoying PCSO might tell you! The Court of Appeal defined a breach of the peace as being ‘an act done or threatened to be done which either actually harms a person, or in his presence, his property, or is likely to cause such harm being done’ – see R v Howell.
Breach of the Peace has been interpreted in court to mean that the cops can use this power in the removal of protesters from private property. This is because the breach of the owners 'property rights' could lead to them using force to remove the protesters!
Talking to Cops: Police often see Breach of the Peace as a catch all law. Explain that case law has defined Breach of the Peace as ‘an act done or threatened to be done which either actually harms a person, or in his presence, his property, or is likely to cause such harm being done’ – R v Howell." Ask the officer how your actions are likely to cause a Breach of the Peace under such a definition. You can then argue (politely) any point they make.