Despite the official advice, there is a way to do it!
Firstly (let me get it out of my system) I was disappointed (but sadly at this point not that surprised) that Inventor would have such a nonsensical quirk as not letting you copy and paste sheets within the same drawing. It's the sort of thing that users would need to do often but the programmers at Autodesk (who seemingly don't actually use Inventor) wouldn't actually implement. Oh yes, there's a work around (copying into another drawing and then back). But that sort of menial task is left up to the user. OK, end of rant. 🙂
So here's the trick. Right-click and copy the sheet, as per normal. Then, instead of right-clicking and trying to find "paste", first left-click on some blank space in the browser panel, then right click. Surprise - the Paste button appears and works!
There's another way of doing it too. Simply do the same procedure as above, but use Ctrl+C and Ctrl-V. You will still need to do the left-click in the browser before pressing Ctrl+V though. The left-click is the trick.
No more creating dummy drawings just to do a simple copy and paste. Inventor, please take note and (in the nicest possible way) get a handle on your own program. 🙂
Kudos to @draftingG29CQ for figuring this one out.
Screencast link: https://autode.sk/421Mos7. This has been tested on 2023.2 and 2023.2.1.