If it's a scam, lying about losing the file is useless, since all the BTC transactions are publicly logged. You can just input their address(es) on the Bitcoin Block Explorer and see if any of them are being used.
Of course, that doesn't mean you'll have any legal recourse against them.
The addresses being used don't necessarily indicate the exchange owner was lying. An enterprising AWS engineer might have been able to recover the wallet somehow. At a value of $200,000+ USD it'd be worth looking for.
Of course, that doesn't mean you'll have any legal recourse against them.