How to have good early dynamic for enemies to lovers in slow burn

I'm writing a story that is balancing a plot around a missing private investigator that the MC is trying to find. Like noir detective, very conspiracy/supernatural stuff involved in the theming.

The love interest, plays the role of the last resort, and she hires him to help investigate the disappearance. He's involved in a lot of ) criminal networks. Basically a very high-end freelancer for illicit jobs, dangerous, high risk-low reward type of interactions.

I'm having a hard time figuring out how to write the first meeting and coming up with ideas on scenes between them that can immediately follow this. How the story progresses is after 4 months of investigating on her own with little success and no progress, she seeks an audience with him to ask for help. They have never met prior to this. The motive for why he will agree is because she has something valuable that he wants, and she has no option but to turn to him.

Its not typical enemies where they hate each other, but their values are so different from each other that they do not get along at all with each other.

I have a good idea of what I want their relation to develop into once enough events happen for them to be more than just tolerant of each other, but I'm really struggling with the first couple interactions.

How should they speak to each other, how can I go from mutual disdain to being non-hostile with each interaction? How do I keep it tense without making it seem awkward?

Do you have any scene ideas I can use to help develop their dynamic?