I'd recommend approaching this as a two-step process of cutting the shape and then preparing the edges. In my experience the cut edges of 5 mil annealed copper (0.005", about the size you're looking at) are always sharper than I want to be casually handling, even without burrs. It's also a bit of a delicate edge to ease because of how malleable thin, annealed copper is.
I suspect die-cutting is the way to go here. Most projects I've seen or done with this material have been smaller-scale, where it's practical to cut one to a few pieces using a variety of hand snips. These snips give you basically a zero-clearance shear and make the cleanest cut (rotary cut-off wheels, especially hand-held, are by far the worst) but the edge still demands attention. I expect this would be comparable to the positive/negative die cutting method you've considered. With a thinner foil, you might be able to get results with a knife edge but 5+ mil is thick enough, and copper ductile enough, that I'd expect more of an embossing effect if you tried it here.
The challenge is that you're going to want very little clearance between the positive and negative dies. For example, using this die clearance calculator suggests 5% of thickness as the "regular" clearance for any reasonable set of parameters (C110 H01 being the closest material option). That would be 5–25 µm clearance on your die. I'm hardly an expert on die-cutting but this doesn't sound like something you're going to achieve in-house.
That doesn't mean you need to resort to specialty fabrication, though, especially if perfection is not the goal. My advice, since your stock is inexpensive, is to make a prototype in-house ASAP with as reasonably tight a fit as you can achieve between your positive and negative dies, using your available tools and whatever method you're most comfortable with. It's very possible that will serve your purpose. (You'll need to keep your negative die aligned on the press, of course.)
Depending on the results of your cutting process, your edges might get rolled a bit. If this happens, I'd try placing the cutout concave-down on a sheet of medium-fine grit sandpaper and giving it a few strokes with light pressure. This might be sufficient, since a workable die shouldn't produce any burrs, and it's "quick and dirty" enough to not be infeasible for a batch of several hundred cutouts.
If you don't get any dishing, that seems like a bit of an issue, as your edge isn't exposed all along one plane. But believe it or not, you can do quite a bit with something as simple as a scouring pad, when working with a material as soft as annealed copper. If you need a bit more abrasive power that will still fit nicely into the nooks of your shape without creating any sharp nicks or dents, try a sanding sponge. If you need to get into smaller nooks or you're worried about bending/crimping your cutouts using a pad or sponge, a small round file can also do the job, though likely more painstaking labor with more aggressive action (you could accidentally cut a notch in the edge).
The possibility that a bit of die-roll would actually help in the edge preparation step might seem to imply that more clearance is actually helpful. To some extent that may even be true; however, the more stretching-deforming action you have along your edge, the less likely you are to get a clean cut without burrs. As the edge rolls, it will become work-hardened. The irregular shape of your oak leaf cutout won't help here, as stresses may be less consistent along the path of the cut, and you may start to move more in the direction of tearing/ductile fracture. So I would still start by trying for the smallest-clearance dies you can quickly prototype.
Then again, it's possible that you'll decide the rough edge that results from your cutting method isn't too sharp for your purposes after all. It doesn't sound like you're producing a piece that will be handled very frequently or by very many people after it's finished and you're going to have to handle the assembly with care simply by virtue of working with such thin copper. Yet another reason to start by prototyping and then address problems as they appear, rather than ahead of time.