You’re going to do one of two things - Union the dog with the entire ornament, OR subtract the dog from just the center part of the ornament:
On the left, I have the whole ornament, grouped together as a single thing. On the right, I have just the middle portion. When filled, it’s a little easier to see why things work the way they do. When doing a union, you’re keeping the overlap of all filled areas. Weld does this too, but it doesn’t know which things you want it to “keep together”.
Booleans only work on two things at once, so it’s very explicit about what is being merged. So, on the left side, if you union all the filled bits together, or you subtract (difference) the dog from the shape on the right, you get these:
