You'll need to use the barrel charger for applying firmware updates. The Lemur Pro has additional circuitry to support changing the battery even if it completely drained.
Wait, ignoring firmware... you need to have some charge in order to charge with USB-C? Couldn't it use the lowest power modes until it has enough juice to communicate? This seems like a serious flaw with USB PD.
That seems to be an implementation flaw. USB-C provides 5V by default with no negotiation which should be enough to power the controller and allow it to negotiate a higher voltage.
> USB-C provides 5V by default with no negotiation
Just to be pedantic, USB-C does need negotiation before it provides 5V, but that negotiation is just a pair of resistors of a specific value.
(Since the USB-C cable is symmetric, this negotiation is necessary to decide which end will provide the power. One end has both resistors wired as a pull-down, and the other end has both resistors wired as a pull-up. The end with both resistors wired as a pull-up will initially provide the 5V power.)
I have no idea. Maybe they can't, but we won't find out for 5 years when someone brings one out of a closet. Maybe there is enough current leakage when in non charging mode for them to run that much electronics.