This response is provide based on the information you provided. You should not use this content as the sole source in making your decision. Seek guidance from your chain of command, JAG, IG., or other appropriate agency. Do your own research. Then make an informed decision.
While you are under no obligation to provide the lease, they may have nothing but your best interest in mind. They may want to check to see if it is a fair lease. They may also want to look at the lease to see if you can get out of it without penalty so you can move back into the barracks. It is hard to know what their rationale is.
Getting an apartment without talking to your leadership might not have been the best course of action. Just because you are now living off post does not necessarily entitle you to receive BAH or separate rations. These are decision that will be based on barracks availability and whether the commander wants to grant you separate rations.
You have to chose if you will provide the lease. To the best of my knowledge they cannot require you to provide the lease. However failing to provide the lease could potentially open another round of demands by the 1SG. Perhaps the best course of action is to be humble and approach the 1SG and/or Commander on open door policy and explain the situation and that you were not trying to do something illegal or behind their back. Usually up front and honest communication will solve issues rather quicly.Hope this helps