I'm running it on stock Android, and it doesn't recognize Google Play services, so it's running a background service. I've read several bugs in the GitHub indicating this has been a problem for years, with no solution. Always marked as wontfix 🤦♂️
Discussion
If it's using Websockets it will hammer the battery as you say as it's running at all times. i wonder why it's not using FCM though? 🤔 Seems odd.
I know on Graphene if you install Signal before installing Google Play Services it defaults to Websockets. If you want to use FCM you must first install Google Play then install Signal.
If you have to use Websockets might be worth trying Molly FOSS to see if it's not as bad as Signal running unrestricted battery.