Is the anvil steel on top, or is it semi-steel, or just iron? That all makes a difference. Also the fact your project is decorative and not structural makes a tiny difference, too.
Truthfully, anything can be welded given the right process. The best way to weld cast iron is with nickel, it mixes beyond the weld into the base by cladding and diffusion making a weld that isn't brittle. It's also possible to flame weld it with cast iron rod, it's artisanal and it works, I have some of that material stashed to pull a hat trick for an antique repair. Or you can just MIG it, the weld will flow just fine, it just won't be strong in cast iron. All of these options involve heating the base metal in a preheat oven slowly, then cooling slowly, over 24 hours if possible. That's if it were a break, crack, or repair.
You really aren't going to hurt anything by zapping a few tacks on there. If it doesn't work, they'll probably just plunk off with a hammer and you can try JB glue instead. Or Permatex Emblem Adhesive, that stuff's sick. Shoe Goo had many successful dives to the Titanic, so that should be good too.