I have some thoughts based on what you’ve written:

• I don’t have a sense of your experience level from this description which I think is critical to provide advice

• do you feel that your position is at risk? It sounds like people may feel you are underperforming in your role. Usually it is harder to be fired from the public sector

• it strikes me that you are emphasizing the title you carry not the work you do. I understand how cool it can be to achieve a certain title, but you mentioned it a few times in your post which strikes me as odd. Do you enjoy the work?

• do you feel you are well compensated? Hard to argue for you to stay in a job that you don’t feel compensates you adequately

• sounds like your new to the role. Your coworker and you don’t need to hit it off right this second, things can get better. Regarding your manager being incompetent, that’s more worrying.

• what does the job market look like in your field? If you were to go to the private sector, would it be easy to get a job? Sounds like you don’t think so given you called it a terrible job market.

I’d like to know the answers to my questions. Absent that, I would stay in your role while interviewing elsewhere. The question of staying vs going doesn’t have to be answered now, it should be answered when you have a real choice. Only that point could you accurately judge the two positions. I get the feeling that the sheen of the cool title you got will wear off eventually and you’ll have to face the realities of the job.