I am not asking about the energy that is actually stored inside the BESS, or the percentage that can be recovered from it. Neither am I asking about the efficiency of inverters, or transmission losses. 

I'm interested in the amount of energy that is additionally needed to operate the system, for example for cooling. Any technology welcome!

I've found the following information online:

A 1.5 MWh lithium battery for research built to minimise cooling requirements, with a claimed average cooling need of 120 W:
https://www.batterietechnikum.kit.edu/downloads/2023_Datenblatt_EN_Li-Gro%c3%9fspeicher.pdf 

An article and advertisement by a company that builds battery enclosures, claiming that lithium BESS produce roughly 1% to 2% of charged / discharged power as waste heat. As an example, a 1 MWh battery being discharged at 250 kW is claimed to produce about 5 kW of heat:
https://www.solarpowerworldonline.com/2019/04/the-importance-of-thermal-management-of-stationary-lithium-ion-energy-storage-enclosures/

This question was prompted by a colleague who mentioned that a specific lithium BESS with a capacity of 20 MWh located in Central Europe would in February discharge at a rate of about 250 kW to supply power for its own cooling even when not doing anything else.

Given the above numbers, these 250 kW seem excessive.