How come Kay doesn't know that a judge's gun is coded to their DNA?

This is set up earlier in the film when they're in the elevator, with Kay stood between Dredd and Anderson.

Anderson: He's thinking of making a move on your gun.
Dredd: Yep.
Anderson: He just changed his mind.
Dredd: Yeee-up.

Which shows that Kay didn't know that it would be a fatal mistake for a non-Judge to try to use a Lawgiver. Dredd knows there is no danger from it (because of the security feature) so is unfazed by the threat.

This is perhaps understandable, given that regardless of how common a sight Judges are in Megacity One it's rare for a Judge to lose their gun and likely rarer still for anyone else to survive trying to use one. So it's very probable that this security feature isn't well known, even amongst hardened criminals (unlike in the Stallone Dredd film where it's known even to pawnbrokers).

Which is somewhat backed up by the point that Dredd specifically makes as they leave the Halls of Justice that losing her primary weapon would result in an immediate fail for Anderson during her evaluation.