Is it permissible for a Muslim man to engage in Mutah with a Muslim woman who is living with a non-Muslim man without being married to him?
Bismihi ta'ala
This Muslim woman should not be living with a non-Mahram person anyway. So, she needs to remove herself from that situation.
As for the man doing mut'ah with her, that's going to be between them, and if they both meet the valid /halal requirements for it.
And Allah knows best