I do that all the times. Haha. One of the definitions for "betray" is "unintentionally reveal" or "to reveal unconsciously."
Someone might have told you a secret. Then in the excitement of conversations, one thing leads to another, and you accidentally reveal it and only recall it's a secret afterward. Unfortunately this happened to me more than once.
My feeling is that intent or (minimally) consciousness of wrongdoing is required at the moment of betrayal; accidental betrayal would not usually be designated 'betrayal' by a native speaker; another term or phrasing would be used. However, others may disagree, and it is of course the case that the Vietnamese 'phản bội' may have a wider semantic field than the English 'betray'.
"cố tình" is not an adverb, "đã" is not an auxiliary verb, and there's no verb conjugation in VNmese. ><
but to answer your question, yes it's a normal structure to have đã + cố tình + [2nd verb].
"đã" is a time marker, "cố tình" is a verb meaning to be intentional in [doing sthg], "phản bội" is a 2nd verb mentioning the action deliberately done.
Let me be more specific: I think that typical use of the English lexeme 'to betray' and its derivatives such as 'betrayal' has a defeasible implicature that the betraying agent acts intentionally, in particular with the prior belief that the betrayed patient, upon learning of the act, would regard it as one of betrayal. This, like nearly everything in natural language, is not a hard and fast rule, but I regard it as sufficiently probable that adding the adverbial 'on purpose' to the present sentence is pragmatically infelicitous in the absence of further context. Of course, others may disagree.