I think they have push notifications in XMPP these days. At least Prosody has modules like mod_unified_push and mod_cloud_notify](https://modules.prosody.im/mod_cloud_notify) and that seems to be supported for example by Conversations.im

To be honest, I didn't have lots of battery drain, back when I used XMPP. And other old-school protocols like e-mail and sip voip don't seem to be very bad either with whatever mechanisms they use. Or my phone isn't reporting battery drain correctly.... And with Matrix I also had to set up push notifications manually, or it'd just receive messages with a long delay per default.