So i am going to make three suggestions.
First you may be able to find a siren that you can trigger a sound on that would act like a doorbell. I don’t know how viable that is, but i know some sirens to have sounds available other then the main siren.
A far more expensive option could be to look at the one Arlo Doorbell kit with 1 camera. It would get you started with the Arlo ecosystem, get you a doorbell with a camera, and a chime i believe. It isn’t cheap, but not priced to bad when compared to other video doorbell solutions. The additional chime can do many things from being a siren, presense sounds and vairous doorbells.
Lastly if you have a always on computer or device that you can turn into a audio device for Smartthings you could trigger the sounds with that. The combination i am thinking of is like Bigtalker in smartthings with a computer or android device that would do the audio playback. There are also some solutions for Alexa and Google Home that allow you to send sounds to those devices. I believe they require a piece of software in between smartthings and the Voice assistant though.
Each of these options can be used well beyond a simple door bell though so those are some added benefits right.