The maximum eclipse any GSO satellite will be in is 72 minutes, per Intelsat. Thus, a launch can't happen that puts the satellite in an orbit with an eclipse period of longer than 72 minutes, or else the satellite might fail. This will be more sensitive for an all electric satellite, which needs to take into account not only the present time but also the future eclipse time. The highly eccentric orbit can lead to longer eclipses than might otherwise be had.
When a satellite is launched, it is in a low power state. There is still battery drain, however, and it might take a while to get to a state where it is power positive. The time is longer for the low power state than the 72 minutes, but it still must be reached. A Geostationary launch will drift for a period of time in LEO to get to the optimal second thrust location. For example, Thaicom was deployed almost 32 minutes after launch. SpaceX also requires that all customers stop charging their batteries 13 minutes before launch. By the time the Solar Arrays are fully deployed, it could be almost an hour without power. If it isn't deployed straight from the rocket (There's a certain amount of rotational velocity that can be imparted), then it could take time to orient correctly. All of these factors add up, making launch one of the key requirements for batteries for a mission!
That being said, the optimal eclipse time for these satellites will be close to Earth, where it will pass quicker. For all-electric, this is even better, as the thrusts tend to be required when at higher altitudes. I suspect this is a limiting factor in most GSO launches. The window won't be particularly tight, but I suspect it is on the order of hours. Depending on the exact alignment, there might be two orbits of a launch window where the requirements are met, IE, launch at 00:00-00:15 or 01:30-01:45 (Numbers purely made up), although that kind of launch window usually isn't published.