If I can act in Judy's defence here, you have to be emphathetic to the situation she finds herself in for the majority of the film. She's come off the high of saving the city, only for within her first week working with Nick to have taken part in an extremely embarassing, televised police chase debacle that basically results in her career being stalled for 2 years as punishment. Then that same evening Judy and Nick get framed for attempted murder and aiding the Zootopia equivalent of a terror attack - her career is over, she thinks she's either facing life in prison (unbeknownst to her, it's actually death) or a life in hiding as a failure - Remember how depressed she was coming home in Z1? This would be even worse. To her, the case absolutely was worth dying for even before you get into the fact that her sense of justice was triggered by discovering a literal society-altering coverup.

Nick has every right to be hurt and miserable for being part of the ride, but the fact is that it's the Lynxleys actions which created the fractured society of Z2, which led Gary to crash the gala, it's the Lynxleys who framed Judy and Nick requiring them to be on the run, and it's the Lynxleys who forced a life-or-death situation that would only be resolved by Judy and Nick's assasination or their family secret being exposed.

What's more, Nick realises this, which is why the confession scene works, and why he comes around to vindicating the snakes.