Maybe we can look into adding this with a toggle ability, which means the default setting in "M" panel would be the original GTA SA siren, and once you change it into the custom siren you're suggesting, the siren you hear on your client will be replaced. This would make it opt-in, so only players who like the custom siren will be hearing it, both while using it (driving a police vehicle) and being chased.
This would implement a way to avoid the GTA siren "giving you a headache" as you said, but not unnecesarily force it upon players who are indifferent or don't like it. Although the option to toggle it already helps a bunch, also keeping it opt-in instead of opt-out is a conscious decision.
I will look into it soon..