I'm not sure if "cruft" is the issue. Or maybe not the only issue. At least for the 2 extra paths which I can see clearly on the photo of the screen, it looks like the Shaper software might be trying to close some open paths. I don't actually know if it would be capable of that, but the placement of (2 of) the extra horizontal lines matches exactly the ends of some open paths.
The issue which I was concerned about, when I first looked at your file, is that you have not only the cutting paths in the file, but you have all sorts of other paths, which have no other purpose than appearances. For example, short paths which create the appearance of toes on some of the animals. Eyes, lines to define legs and knees, etc.
So I would probably make a 2nd layer in the SVG file. On one layer would be only the cutting paths. And the other layer would contain all the extra paths which are needed only for the appearance. Hide that "extras" layer when you send the file to the cutting software, so that it can only "see" the cutting paths.
Because, how does the cutting software know which paths you want to be cut out? Isn't it going to try to cut them all?
To do that, you'll have to use Path menu > Break Apart, to break that large compound path up into its individual pieces. You might even have to break paths at certain nodes, to accomplish that completely. It will probably take some work. Let us know if you need details for that part.
Oh yikes! I just noticed that you have anti-aliasing disabled. Is there a reason for that? I don't know if it will affect the cutting of the paths....I guess probably not. But you'll have a better quality experience visually, if you enable it. File menu > Document Properties > Page tab, near the bottom, Use Anti-Aliasing. You'll have to dock the dialog to be able to see it (if it's not already docked).
(Actually it probably is already docked, because that option is always enabled by default, so you must have disabled it, at some point, on purpose.)
(I wonder if aliasing would be considered "cruft"?)