You definitely don't need the app for that. The solution @ogiewon linked to above (here: Ring/Amazon/Alexa - who needs IFTTT? Not me) has a way to do this with just Alexa and virtual devices from Hubitat, and as mentioned, you could then use any automation (such as something in Rule Machine) to manipulate the Dome siren. The first part might be a tad more work (or not--you'd get to skip all the custom apps and drivers), but you avoid custom code on Hubitat (I'm really not sure why the app makes you do some of this yourself and still passes a lot through Alexa when it looks like it could do a lot on its own, and I just looked through the code am not sure I'd personally run it on my hub--appears to be using a lot of "regular"/synchronous HTTP calls when they could be asynchronous and not cause the app to stay awake and wait for a response, no disrespect intended for the author who took time to write it).
The second thing--making the Dome respond to a doorbell button push--is something you'd have to do either way. The custom app you originally found and the workaround above are both just ways to get the doorbell into Hubitat; automations you create using it are still up to you.
