is it legal to put Grotesques in a Cult Raider?

The idea is to use red Grief Raiders to advance and charge, before unloading the Grotesques in the next turn.

Another question: If it is legal, do you think it would be cheesy?

It is legal. I don't think it is cheesy. It may not be super effective.

I run all of my vehicles as a Kabal. Primarily so I can keep track of which one is which. They're all similarly painted.

Certainly legal, and make sure that if you're going to get a charge off on T2 to use the strat to bump the PfP chart for them!

Also, be aware that if an enemy can surround the raider and then blow it up, you're likely to lose the entire unit since they can't disembark.