What you’ve presented is actually quite a dilemma.
To answer the second part first: no. There’s simply no way to determine if your files have been copied – at least not in any way that absolutely says they were copied with malicious intent.
The problem is, there’s really no fool-proof solution to your scenario. In fact, I’ve heard of companies occasionally electing not to repair a hard drive, because it meant that sensitive data might be visible to repair technicians.
Your options to secure a hard drive are limited, but if you can plan ahead, there’s a chance.