I don't have a name for that little piece, but we all know what it is. The piece comes out easily on both my Gladiators and one of my Victory-class. The base support doesn't fit without it... but with some filing or trimming, I think it could. Has anyone eliminated these pieces and just made their ships fit on the base supports?

Why would you? They make the ship much more stable on the base and you can't see them when the ship is on the table.

I just glued the one I had come off back in with some super glue.