Yes. 'Had better' is a set phrase meaning 'We would be wise to'/'It would be to our advantage to'.
Rover
Does 'better'' have an adverbial role in this sentence?
Yes. 'Had better' is a set phrase meaning 'We would be wise to'/'It would be to our advantage to'.
Rover
And ''over with'' is a word separately or integrates ''this'', as it was said ''get over with this''. Thank you.
'Get this over with' means 'get this finished.'
Your thread title means 'We would be wise to get this finished'.
Rover
But, could I had written "we had better get over with this''?