Is "bettern't" an OK word to use?
How about, to sound more formal as you said, you say, 'You had better study, hadn't you?'
I've only heard this word widely used in Devon, England. I was living in Plymouth, and people say bettern't you, bettern't he... all the time. You get used to it, and I always consider this is part of the Devon dialect.
edit: I was living in Devon between 1983-1995.
Here is a link to the transcript of a recording from a BBC programme discussing Devon accent and slangs, the original page was archived and recordings deleted, so only the transcript is available. https://sounds.bl.uk/related-content/TEXTS/021T-C1190X0013XX-0501A0.pdf