Hi,
In the first sentence, I'd use "I've an idea", even with American English, because I'm picturing the person saying it
very quickly, she has just thought about it, and she's saying it with a particular intonation.
In the second sentence, i've got is better because they person is saying it more slowly and with a different intonation.
I've seen i've with a. english lots of times.
That's just my opinion and how I'd use it.