#Yes, you can cast spells through your illusion.
This question is really about 2 things: does Forcecage block Channel Divinity, and does it block the magical connection between you and your illusion?
Forcecage is very specific about what it blocks: matter, spells, teleportation, and ethereal travel. So things like sound, vision, and temperature still pass through the barriers of the cage, because we have no reason to assume they would be blocked. So because Channel Divinity is not a spell (rather, an action that you take to channel divine energy), it can be cast through the Forcecage.
Similarly, Forcecage states nothing about blocking the connection between a magic user and an ongoing spell effect. For example, you would also be able to concentrate on a spell that you previously cast, taking effect outside of the Forcecage. And looking at the text of Invoke Duplicity, there is no reason to assume it would be blocked either:
For the duration, you can cast spells as though you were in the illusion’s space, but you must use your own senses.
So as long as you can see the place that you want to target with your spell while you're inside the Forcecage, you can cast spells through your illusion standing outside of