I don’t own a la cloche. However, I make challah bread from scratch every Friday. I usually use a silpat on a cookie sheet, but I’ve also used parchment. I have found that the parchment shifts as the bread dough expands during the last rise and during baking. This is not a serious problem, but I prefer the heavier weight of the silpat and the fact that it doesn’t move.
Why don’t you try seasoning the lid of of your la cloche? By that, I mean wipe the inside of it with vegetable oil. This is only a suggestion. Maybe somebody with more la cloche experience can chime in here and tell me if this is a good idea.