Conversations can be a unified push distibutor: https://unifiedpush.org/users/distributors/conversations/
..and I'd trust it (battery-wise) with that. I have an old tablet with conversations running without battery restrictions on it, and if I'm not actually picking it up and using it it regularly goes 1-2 weeks on an 80% battery charge before it dies, the whole time giving audible notifications for XMPP messages/calls (which I attend to on other devices).
No, the "distributor" is the part which runs on your portable device, receives the push notifications, and wakes up the target apps as necessary.
https://unifiedpush.org/developers/spec/definitions/