I am concerned that if I lever them off I will either snap the cast metal or damage the veneer.
This is a valid concern and honestly I don't know the best course of action here.
To help put your mind at rest about part of it though: since you're planning on painting that means you can fill any defects in the surface that might be created without them being evident, as they usually are where a clear finish is used. Done properly fills on wood are truly invisible after paint goes on.
Trying to save the ornaments
Honestly I don't think there's a reliable way to get these off and ensure the ornaments don't get damage. These tend to be a cheap metal (often zinc) and it's brittle and soft, so takes damage easily. But you can try a trick using a modified putty knife as the levering tool. IME it's not 100% reliable although better than some other methods.
What you do is take a common putty knife and sharpen a slight edge into it, not enough that it's actually sharp, just enough that it helps it slip underneath the edge of the ornaments. Go for a single bevel, like a chisel, rather than a typical knife edge.
Then round the corners. Having rounded corners greatly helps reduce marking of the wood as you're levering up the applied decoration.
In use the putty knife is positioned at the edge of one of the ornaments, at a shallow angle to the wood surface, and wiggled under or gently tapped with a hammer/mallet/spare piece of wood until you have some purchase and then you start to lever up. Once started you might like to slip a piece of cardboard underneath to further help prevent marking of the wood.
Best of luck if you try this!
Ornaments are expendable
In this case you can just dig into them, then using a pair of pliers grasp the nails and pull them directly out, twisting and levering as needed.
You can use any tool at hand to help expose the nail heads so they can be gripped. A narrow chisel, the pliers themselves, even a steel barbecue skewer could be useful here.