I was going to say that Peggy's also a nickname, but you meant Daisy being linked to something else, not another name for Margaret.
I'm not aware of Daisy being anything other than a name in its own right, and had to look it up on the internet to see that it was a nickname for Margaret. It's because the french name for Oxeye Daisy, Marguerite, is also the French version of Margaret.
Would Marguerite be better than Margaret? Why can't you have Daisy as a name in its own right?
My son's neurologist's first name is Kindra. I think it's a bit childish sounding, not a 'serious' name, but it didn't stop her becoming a neurologist. There's another professional (albeit in the entertainment industry on the production side) my DH has come across through work, named Pixie. She's one of the higher-ups, so really I don't see any name, unless it's utterly ridiculous, stopping somebody becoming whatever they want to be. So I am in the camp of Daisy standing alone just fine.
If I wanted my dd to be called Daisy, then I think I'd have to call her Daisy. I mean, it's not (or wasn't!) a name I'd choose, but these days it does stand as a name on it's own and I think it wouild cause confusion if she was Margaret on official paperwork, but everyone called her Daisy.... it's not like 'Maggie' or something that is obviously likely to be a nn for a Margaret.