I used the default vehicles. The only mods that I had were LCPD:FR Mod, A custom police hat, custom sirens and horn sounds, a blood mod that adds realistic blood, and a first person mod. I still encountered the problem, I don't know what is causing it.
Edit: I am now sure I am having the same problem as Tom H, as he posted above. The only solution to prevent this is to not do traffic stops at all, which I really don't want to do.