They probably won't... Links between satellites will use a different routing scheme. Traditional network protocols will "disappear" at the sending ground station, only to "reappear" on the receiving one.
A video link of a simulation was posted on a different comment that shows the "routing" between multiple satellites.
Why wouldn’t they use something like BGP to route between the satellites? Because the topology is explicitly known ahead of time and so the routing can be more “hard coded?”
An interesting side effect of this network is that latency experienced by any two users, anywhere on the network, should be close to equal.
A video link of a simulation was posted on a different comment that shows the "routing" between multiple satellites.