What lentiviral vector are you using? Have you consulted the vector literature to see if your problem is included in the troubleshooting section?
Explain how GFP is not inducible.
Edit - I just reread my post and it sounds confusing. Yes, you confirmed your insert was present in your vector via sequencing and propagated that plasmid. You transfected this purified plasmid in your 293T cells, so there should only be your GFP tagged construct. I am confused to how you are going to sort out your cells. The presence of the GFP will indicate that your GFP tagged construct is 'leaking." This could easily be confirmed via western blot to show that you have one band, indicating your tagged protein.