Yup! No reason not to give it a go. Good paint in a can, or go buy a spray gun and practice on some junk first. It's a skill and toolset you'll appreciate having in the future. If you can paint a fence, you can paint an engine bay. It's not hard to DIY the internals. Just be sure to keep track of color codes and product used so the body shop can choose their steps wisely when finishing.
Learning to fit sheetmetal and remove dings and dents is a whole other level, and not something everyone is comfortable learning or doing. Pros know how to get it done and done proper. But all the stuff you don't see from outside is fair game IMO.
Also, strip and repaint or seal and repaint over the copper first. If you don't, normal body flex is likely to crack your paint at some point and eventually you'll be under the hood and seeing copper through cracks, chips, and rub marks. Copper will show super obvious through turquoise, so I wouldn't even entertain only painting after assembly.
