I'd rather not give them any useful information unless it's something inherently harmful to them.
Like if we could hint at something about unbinding Akuma to sow discord in the ranks. Not sure if we can actually do that, but that's the level of disruption I'd prefer.
If we really want to screw them we could use our Crown questions to try to design a more reliable akuma detection method and sell it to the Jade Court.
Knowing it's a trap doesn't actually stop it from being a trap. Putting ourselves at a disadvantage is a bad plan, even if we can probably work around it.
Well, CCC makes traps of many kind into 'I'm not trapped inside with you...' scenarios.