The battery consumption of Conversations with or without FCM is about the same. (Battery consumption in both cases depends on a lot of factors like frequency of messages, quality of network or even how often your operating system kills Conversations due to a lack of memory. But eliminating all those factors battery consumption is in the same ballpark. The only difference from an end user perspective is that Conversations doesn’t have to ask for a Doze exemption if you use a server that has the push extension.

Conversations.im does just that, uses FCM to get the messages in the background. They recently open sourced push server that interacts with FCM: https://github.com/iNPUTmice/p2