Green Cards / citizenship and the TSA have nothing to do with each other. All foreigners coming to the USA for aircraft specific training need to get a TSA approval.
It absolutely true that you do need your TSA approval before you start your sim sessions.
Dont see the link to a green card though.
They have every to do with eachother if you work for a US 121 carrier. The TSA requires each carrier to ensure not only that all pilots have the right to legally work and live in the US but also furnish said proof to the TSA, so that they can verify it.
I just had to talk to the person at my company who deals with the TSA and he verified that particular point.