I'm trying to describe being on top of things accounting-wise but can't quite find the words. Is there some kind of idiomatic antonym for "cooking the books"?
In colloquial use "keeping clean books" is a phrase that is often used and would be widely understood.
(Technically, you could say "following generally accepted accounting principles" or "using GAAP" but that's only idiomatic in accounting and finance.)
to keep impeccable books
Legitimate book keeping? It's clear, concise and rather straightforward. However, another possibility (although very informal) is to refer to the book keeping process as being on the up and up. I read on a popular blog, 'The Grammarist' that this idiom's exact origins are mysterious, dating back to the 19th century, likely American and it appears to come from sports betting. Although it has another meaning - on the rise - the blogger states that this usage is rare and examples are difficult to find.