I think I would have pulled the cork when I inserted the caged queen.
The candy cork, not the open end, of course. I've laid the cage flat, screen side down, with some of the open end screen over a frame top, so the queen can retreat from any initial "mob", while still having
some direct contact. You can also strategically place the cage so the candy hole end is visible through the inner cover hole. That way you can see what's going on without popping the inner cover more than once (to remove it).
The cage will lay flat under the deep side of an inner cover, and with a new package not be too extensively filled with burr comb in the first week after she's out (allowing her a few more days to settle in after before you disturb the hive again to pull the cage). 'Perhaps a bit over cautious, but it's simple method with little downside.