How can I inject a taxonomy menu as the sub-menu of a standard menu item?
Option a) Unless I am mistaken, can you not just pick where your Taxonomy Menu will be parented (Menu Location) when you create it? The only problem I think you might get with this is I think it requires you to rebuild the menu every time the taxonomy is updated

Option b) I have done something reasonably similar to this on my site, I used Menu Views module, to allow me to put a view into my primary menu.

If you create a new view which lists all of your taxonomy terms, as a block display, apply any additional filtering/modification in the view if necessary (you may not but in my example I wanted to). Then when you add a new link to your main menu, you can select to use a view rather than a link. From there you pick which view you want to use, select the 'Category' menu item as its parent and save (after saving you then have to enable it in the menu, which appears to be a bug with the module itself)

I can't guarantee that this is the best solution for you, but it worked when I just tested it