No. Taking the wakelock is significant battery drain. Because of poor mobile OS architecture you have to take the wakelock to leave connections open. On normal Linux you can just have the OS suspend anyway and most of the time the connection will be there when it wakes up and if it doesn't the client automatically reestablishes. (this is exactly how the normal push connection on android is maintained anyway, there's almost no real overhead to having a couple more.)

Mobile push is a combination of brain damage and bizarre propaganda targeting developers.

> Network activity is quite significant for battery drain

It's not activity itself but what app does in background in reaction to incoming packets. Clients I used (Bombus and Xabber) didn't support that XEP with no problem. App may just ignore some stanzas when not needed.