Run all the pool of prospective jurors through the standard TS clearance deal [reasonably painless, done it more than a couple times myself - but then I have had the FBI up my ass over stolen documents already, since neither mrAru nor I had them, having the phones tapped and a crew surveiling us from next door didn’t bother us, and we did send the guys across the street pizza more than once and set our own camera up watching their house … =)]
Once the jury is empaneled, swear them in, hand them their clearance and tell them that if they open their mouth about anything, they can go to jail. Problem mostly solved.