I'm an assistant prof. in a public US university. I have to go up for tenure this year. The department I'm in is somewhat mismanaged, with lots of politics and favoritism. Many have left the department in recent years, and I don't see any motivation for the chair or the dean to improve the situation. Hoping for a better position, I've searched for other TT positions this year and received an official offer from a private higher-rank university with a much higher salary than what I earn now (a dream position for me). However, unfortunately due to personal problems, I cannot accept that offer and have to stay where I am.
My question is: can/should I use this new offer to negotiate my salary in my current department?
Given that I have to go up for tenure, I'm afraid that showing that I have an offer harms my tenure and evaluation. I'm afraid that the chair, dean, and my colleagues see this as an "unfaithful" act or, thinking that I may leave sooner or later, don't advocate for my tenure as much as they can. Additionally, since the department knows about my personal problems, they can easily call on my "bluff" and be pretty sure that I won't be able to accept this new offer and leave the department this year. So, it could be an easy "no, we can't give you a raise" for them. On the other hand, I don't want to take this chance of having a better salary from myself, if there is not much harm in asking for it. Plus, I think, this could potentially show my worth to my current department.
In this situation, would it be wise to ask for a salary raise? If yes, how should I approach it?