As a team leader, would you file a lawsuit against your company for wrongfully terminating one of your team?

NO I WILL NOT FILE ANY SUIT. BUT I WILL ASK THE COMPANY TO RECTIFY THE FAULT.

In legal context, not anyone can file or sue without there being any of their right got infringed or deprived. so Locus standii is the concept very much important in filing a case or suing the employer. so one who deprived can very well file a case against or sue the employer for seeking protection against the deprivation of his/her rights or for enforcing his/her legal rights. so not everyone can sue or file a case against the employer unless you have a locus standii. My 2 cents. Thanks.

Being frank and straight forward.

No. You might be risking your job, irrespective of the judgement you will become a marked employee.

But you can support, you should first check your company policies and then consult a lawyer and take precautions like consulting your employer before proceeding.

Instead of filing a lawsuit try to get the matter solved by communicating with the team member and management as a representative of either.(With good planning)