You’d better bloody well tell them you’ll need to discuss it with me first.
(The Casual Vacancy, by J. K. Rowling)
Whenever I saw a bare-infinitive follow close behind ‘had better,’ I thought ‘had better’-plus-verb is a loaf of meaning bread. But as you see, there’s a ‘bloody well’ intervention. This gives me this impression that the construction might be a combination of a matrix and a conditional clause - I mean two meaning-sub-breads. That is, ‘You had (in here, there might be an insinuation of ‘a treat, a situation, etc.’) better bloody well, (if you would) tell them you’ll need to discuss it with me first. As a to-infinitve can deliver a conditional meaning as in: People might take you for a girl, to hear you sing.
It’s hard to believe there could be any account for this in any grammar books. But do you perchance read the way I said?