Can you (as a druid/trickery domain cleric) invoke duplicity, wild shape, and then cast spells through the illusion?

I am Dming a DnD 5e game. I have a player who is trying this out. My thoughts are wild shape says you can’t cast spells, and tricky domain says you cast spells as though you were in their space. So RAW it seems you can’t. Their argument was…

I was thinking you can’t cast spells because you don’t have hands or a way to do the verbal components, but since you control the duplicate at will, then you should be able to have IT do the Verbal/Somatic stuff.

Thoughts? I’m pretty sure RAW its not allowed, do you think it would create an unbalance to allow it?