I wonder if this can be fixed by changing both to right.

No, it can’t. The problem is exactly that the right side is too long, so clear:right would still push down the button, and with that, the category content as well. I added two screenshots to the task description, hopefully they make the problem clearer. (Note that I used Save full page in Firefox to make the screenshots; the visible part of the page ends on my screen between the coordinates link and the Authority file heading of the infobox, everything below that is invisible without scrolling.)