He could have said "Except for telling you that I have money, which I don't" (the word "have" would be understood). "Haven't" has become a less popular construction since that film was made. You can still use it, but most people nowadays would form the negative of "to have" by saying "don't have".

'Haven't' is from 'have/has got' e.g. I have got long hair / I haven't got long hair. You can also say 'I have green eyes / I don't have long hair'. Both are correct, you can choose which one you use. As Douglas said 'have got' is maybe less common now. I'm a native speaker and I use both without realising or knowing why!

I think this is just a matter of style. The use of "haven't" is a bit outdated, and might be associated more with an upper class/snobbishness.