ANY time a supposedly insane person claims they were misidentified, it should AUTOMATICALLY trigger a detailed investigation, and I don't care at all if that is burdensome to the hospitals or police.
(To be clear, even without such a policy in place, a large number of criminally-insane people already do claim to be misidentified because “they’re not John Smith, they’re Jesus Christ!” And these people do tend to really believe that, to the point where their behaviour isn’t all that different from someone who’s been misidentified. It’d be very hard to define a policy that would precisely delineate which stories should be looked into, vs. which are balderdash.)
We need more simple automated cross-checks to decrease false-positive rates. We need scalable solutions.
I think you're overestimating the number of people incarcerated for criminal insanity and underestimating the number of resources available.
But maybe the GP meant a not-so-detailed investigation. A regular investigation, per se. A cursory investigation, even.
... but we have the resources to do a security-clearance-style background check on every individual that is incarcerated against their will for reason of mental illness. I'd consider that sufficient.
If the state (or other organization) doesn't have enough resources to do a detailed investigation, then it should not be taking away freedoms for whomever it is unable to perform the detailed investigation for.