Need help perfecting Siren Audio Effect

Hey everyone, I am developing a firefighting simulator in Unity: https://www.youtube.com/watch?v=yJsqRaB5V78

One of the features is a fully dynamic siren that can be winded up and down by the player. We are attempting to recreate the EQ2B Siren. Now we have gotten extremely close and have basically perfected the timings and pitches however there are some small details we still need to be able to figure out how to recreate to make it perfect, and thats why I am here on questions asking.

Our siren works off a recording of the siren peaked at its highest pitch and then with a script that tone is pitched higher or lower at different timings and limits.

Here is a good idea of where we are at: https://youtu.be/2JChh9zsDkI?t=8s

Here is the actual siren: https://www.youtube.com/watch?v=fmanO5I-xLQ

And actual siren in action: https://www.youtube.com/watch?v=7Ff5U0gX454 https://www.youtube.com/watch?v=AQHr5ctdDF4

We need someone with an ear that pays close attention to details and there is a certain effect we are really trying to pin down that really makes this siren stand out. It is a certain sound you can hear when it winds up that makes the sound and we haven't figured out how to approach it, whether this involves changing the sound we use, adding different filters or audio effects, were not sure but we would love some advice from other developers on what we can add or change to make this siren be spot on. We greatly appreciate and are willing to pay for any help as well if we are able to get results.

Rafael Fernandez FlipSwitch Games