To even begin our film pitch, we need to be able to know which company would best suit our idea for our film.
Genre Films; a no-go

At first we considered Genre Films, but seeing as it’s such a big company, there could be little we could do in terms of freedom and how we can represent our detective. Other mystery movies they have made either don’t really follow the “cozy detective” feel we kind of want, or has basic representation and has kept it as its always been, such as Murder On the Orient Express. The mystery movies they produce are typically associated with the action or horror genre, such as Logan, The Martian, and Fantastic Four. While that movie is close to the plot of what we want, seeing the production budget and the company under it, it wouldn’t be a very good fit for our idea. We would want a little more creative freedom since we do have a lead female detective and we are still in the age of Sherlock Holmes and the males dominating the cozy detective type of mystery films. Genre Films won’t be for our film this time.
Next Wave Films; our choice

Next Waves Films hasn’t produced as many films, but the ones they have include mystery films with the simple, sort of feeling we want to go for. It’s not a “cozy detective”, but it’s a feeling of relaxed mystery and there isn’t any extreme action involved in the movie too much. Such examples of this are Following, by Christopher Nolan and Pi by Darren Aronofsky (a personal favorite). Next Wave films has also had films that have included good representation of minorities. For example, in Southern Comfort, by Kate Davis (a female director and producer), the main character is a transsexual male living in Georgia and dealing with ovarian cancer which won’t be treated by the 20 doctors he’s visited. With this kind of good representation and even a female producer, sticking to Next Wave Films will do us some good. Next wave Films is an independent company, which would require a small budget. We wanted to go for a low budget movie so we don’t have to spend much and will be able to have a sort of challenge with how we are able to shoot our film. The demographic for our film may be slightly strayed away from a production studio they are unfamiliar with, and we would have to be creative with our marketing strategies since we have such a low budget, but it is not impossible to earn our money back. Our demographic of an older audience would be drawn in by the idea of a mystery, and could even draw in more of the female audience with their gender being represented well.
Sources
I linked various movies and production companies where they are mentioned.