Hello, I'm Augusto from Boardible, thanks for pinning us here! We do have plans to make it open for game designers to make their own games in our platform in the future, but we are still far from it.
But answering your question, the nice thing about turn based is that you don't really need everything to be synced all the time. So it can be really easy on your network just by sending which actions each player took on their turn. Simulation can be all done in each client device.
Just be careful to sync the random seed at the start of the game (and only use the synced random on things that happen the same amount of times on each client. Leave animations and other visual random effects, with a separate random that is not seeded).
Another thing that you need to consider is how are you going to handle reconnects. You have 2 options. You either have a way to query every state of each component, parse it and send it. Or you can have the actions saved and "replay" the game on re-connection. The first option is faster to reconnect but the second option is easier to implement and has the benefit of being able to replay a given game for debug.
With all this info in mind, any kind of messaging system will be enough for your application, especially if you plan on doing this a hobby. I would prioritize something that you are already familiar and its easy to implement. Websocket seems like a good and easy option, and you can easy scale by adding another server if needed in the future, since you don't need every user on the same server, just need every player in a room.