The first step for me is to be able to understand what you want to do. I didn't quite catch on from your description - well not clearly anyway.
Ok, so the badge is going to be a wood inlay, inlayed into other wood. Right? Remove the stars from the flag, so that the "field of blue" will contain the badge instead of stars?
I'm not clear what you mean by "outlay".....unless the outlay is the piece of wood which holds the inlay?
By "badge union" do you mean that you want to make a line drawing of the badge, and combine it into the field of blue? So that the badge won't be green and black, but it will be just a line drawing? The reason I'm thinking like that, is because these are cutting lines, right? I guess some cutter/plotters need the paths to be certain colors, but otherwise, color is irrelevant. Right?
In one of your comments, it sounded like you wanted to remove all the stars. But a different comment sounded like you just want to remove some of the stars - where the badge is, I assume?
If I have a correct general understanding, here's the first tip I can give you. I can't think of any way to "union the badge" in one step.
Most of the path operations (booleans) won't work with the tattered flag design, because it's too complex. If you want to try anyway, just to see what happens (or prove to yourself) you'll need to combine all the parts of the flag, instead of grouping. Also, duplicate the green badge path first, because it will be sacrificed during most of the operations. Or if you make an outset of the badge, duplicate that first.
The only path operations which might be helplful, are Division or Cut Path. Experiment with a couple of simple paths first. And you'll need to do some node editing probably, too.
Last tip: Make sure you don't have any leftover bits and pieces from previous work or experiments, hiding anywhere, which could cause problems.
Ok, well I understand what you mean about learnig to do it yourself. So I tried to give you just some general guidance. But I can be more specific, if you like. Or you could ask whatever specific questions you have.
Oops, I see Heather just posted while I'm typing. But I'll post anyway.