It's not important if they are interchangeable where you are from. The only thing which matters is if there are different translations for them in German. The expressions "gerade jetzt" and "momentan" are interchangeable to some degree in German, too. But the one translates to "currently" and the other doesn't. ;)
The confusion arises from the way "have" commonly occurs both as a main verb and as an auxiliary verb. As an auxiliary it handles negation and question formation without needing a "do". In the role of a main verb you have to 'support' the "have" with "do".
The given sentence uses "have" as a main verb, so the negation needs a supporting "do".
At least that's the grammar rule I've learned many years ago... However, since I am not a native speaker, I cannot, of course, speak for contemporary colloquial use.