The external battery isn't necessarily your only option. Alternatively, Shimano sells an external holder for the internal battery, the SM-BTC1. So you could buy an SM-BTC1, a BT-DN110 internal seatpost battery, and an SM-BCR2 charger. Now you have a system that mounts externally, works for both charging and wired firmware upgrades, AND could be moved to an internal seatpost setup in the future without changing out any hardware (except cables).

That said, I used to have the exact external battery/DN100/R8050 setup you are looking at, and it definitely worked perfectly with synchro shift. I had a wireless unit in the system but I don't think that matters.

As for "e-tube" compatibility, they are possibly talking about compatibility with the e-tube Windows app over a USB connection. On internal battery systems, the SM-BCR2 charger can also be used for limited diagnostics and firmware upgrades, so maybe that works for external battery systems as well? I don't know if the SM-BCR2 can actually charge the external battery though. I've Googled this and couldn't find any conclusive first-hand reports of success or failure. If it does work, I would recommend getting that instead of the normal external charger, for the connectivity checking and firmware upgrade capability.