So I have a pair of sideskirts off a Z-24 and plan to mount them to the Hawk, but I never knew the “Z-24” logo was more than a sticker—it’s sort of embedded in the piece. Now I figure I can get some sort of body putty and fill that in and smooth that out so it should look fine, but what should I get/use as this isn’t metal? Any particular grit sand paper/special bodywork sandpaper?

I have a friend that has done the same thing you are talking about and it looked really clean, I was around having a few beers and watched him doing most of it.
He stripped all the paint in and around the recessed area first,roughed up the recessed surface with 180 to help adhesion then applied "Bondoglass" a high strength fiberglass filler in two separate layers, smoothed out with 180 up to 240 grit.
Then he finished it off with lightweight auto body finishing putty and finally sanded with 420, primed and sealed, blocked the panel then put on the color.

Wayne Schiff wrote: ....but I never knew the “Z-24” logo was more than a sticker—it’s sort of embedded in the piece.

I have a set from a Z coupe and they have two nuts holding the emblem on. My '87 Z that I had was the same only the emblem stuck out on it that year.

Talk to a body shop and see what they recommend. I don't know why I didn't get into body shop when I was in highschool . I have so many ideas bouncin' around in my head and no talent to put them to use.