Dizzy is referenced with the MC's lawyer - referring to the unknown third heir, not Lily. AFAICT, the only person who has called Lily "Dizzy" is the MC, and he didn't know her growing up. So I have no idea why he picked that name, unless it's just to scare the audience.
Personally, I think it's all but impossible for Lily to be the gold digger. Unless we think Donald was fooling around with an underage girl, the affair could only have been within the last few years, and he somehow managed it without the MC or Katie/Debbie noticing. That's a stretch. Moreover, the Lily we know wants to be respected for more than her looks, has a hard time focusing on practical matters, and doesn't use birth control. None of that seems compatible with being Donald's secret lover. Unless we assume everything we've learned about Lily has been a bald faced lie, it's hard to see how she could fit the role.
I think it's much easier to assume Lily is innocent and the Dizzy 'nickname' is a red herring. But I'm admittedly biased since I adore the Lily we know.