The Starship, BFS vehicle has almost SSTO capability, with very little payload.
The less than SSTO mission profile of Earth To Earth suborbital would likely allow single stage operations not requiring the booster at all.
There will almost certainly be three different Starship models before Earth to Earth is considered.
- Cargo (Satellites)
- Manned (Orbital, and Mars missions likely will differ as well)
Thus the E2E vehicle will likely be a variant of the orbital manned design. Most of the changes will be internal. 200 passengers (or whatever number) versus a smaller crew, but with airline like seating. There will be a simpler life support system, in some ways, and more complex in others. More people to support, but for much shorter periods of time.
SpaceX has yet to explain how they plan to handle all these variants. Time will tell.