As far as I've understood, "bitte" has kind of always a "positive" meaning. And you can actually use just "danke" to mean "nein, danke", if the context is clear. Someone offers you coffee - "bitte" would imply you want some and "(nein), danke" that you don't want. A bit confusing for English speakers, I know. So "nein, bitte" as meaning "no, thanks" would be a bit mixed to a native. To their ears it could sound a bit like "no, (but I'd like some) please" and they would probably ask you again.
There are some exceptions though, like "nein, bitte nicht" "no, please don't". But other than those I would say it is safest to just say "bitte" in the positive "yes that would be nice, thank you" meaning, and save "danke" for the polite refusal - and as a thank you after somebody already poured the coffee.