Iain,
The Offset Path can sometimes move in mysterious ways, but often behaves well.
However its behaving well relies on a good quality/simplicity/efficiency of the original path(s), so a cleanup could be a good way to start; the screenshot by Monika shows a few flaws.
Such Compound Paths can be made from paths that were originally stroked and then outlined and assembled with Pathfinders (or Shape Builder/Live Paint, and thereby have constant distances suitable for Offset Path and/or 1 step Blends, maybe using both to create new stroked paths to assemble, or (literally) pieced together as described by Bobby.
The difficult part is obviously the Compound Path forming the wings/body/head.
Hopefully, you can work with Offset Paths most or all of the way, or at least get some initial parts as stroked paths to work with/use as guides as follows, after ungrouping from the separate parts, and using stroke/nofill paths as the the 2nd image, Smarts Guides being your friends:
1) Go through the paths and clean up/improve shapes, especially at corners, working with tools like the Remove Anchor Point Tool to get rid of funny extra Anchor Points and/or the Add Anchor Point Tool/Direct Selection Tool to reshape parts;
2) Select the Compound Path and release it;
3) Select the innermost path forming the wings/top of the body and apply Offset Path with Offset equalling half the distance to the corresponding parallel outer/inner paths to create the crucial first simple full stroked path; hide the original path;
4) Select the outermost path forming the outer shape of wings/body/head and apply Offset Path with the same Offset as in 3) only negative; hide the original path;
Now you have the two basic (Offset) stroked paths that form crucial parts of the bee; you can get rid of the parts of the the outermost path from 4) that run parallel to the innemost path from 3) as follows:
5a) Cut the outermost path from 4) at the corners where the head meets the wings and the corners where the wings meet the tail of the body; delete the paths that form the shape of each wing (more or less coinciding with the corresponding path from 3));
5b) Snap the end Anchor Points of the head and tail parts from 5a) to the path from 3);
6) Delete the superfluous paths that forms the inner shape of the head, as well as the midmost inner path of the body, two in total;
7a) Apply the same (positive) Offset to the two remaining inner paths, one forming the tail and the lower stripe, the other forming the upper stripe, remember to have a sufficient Miter Limit for the tail path as mentioned by Monika;
7b) Cut the two paths from 7a) in the same way as described in 5a), and delete the paths more or less coinciding with the paths formed in 3) - 5b);
7c) Snap the end Anchor Points of remaining paths from 7b) to the path from 3) - 5b);
Now you ought to have the set of stroked paths corresponding to the Compound Path;
8) Group all the paths formed up to 7c);
You can work similarly with the separate paths, and Group all the paths forming the bee.
The description above can be adjusted depending of which paths have the best quality, and (parts) of paths can be redrawn, maybe using (some of) the offset paths as guides.
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