Sorry to misunderstand your question, if French is your first language, could you post in French and English.
Is "BBB" a BeagleBoneBlack (non Wi-Fi)? What is "BBB" abbreviation for?
How is the battery connected?
Does the BBB recharge the battery?
Does the USB Wi-Fi work when the BBB is powered through the USB PC MINI connector?
If not, it might be using too much current.
I do not know any USB devices that work below 4.4V. I would think, USB devices need > 4V for its own 3.3V supply.
The BeagleBone BATTERY Cape, supplies 5V to the DC in jack through expansion connector P9.5 and P9.6. The BBB should work like connected to 5V power supply to the 5V DC in Jack.
You will need to use alkaline batteries (cells) or rechargeable NiMH to get enough current; (the cape has the batteries connected in parallel, AFAIK you should not do this; the Tesla battery pack has each cell connected through PTC resistor/fuse). For the BeagleBone and USB Wi-Fi dongle to use 500mA, the batteries would have to supply 1A.
Perhaps, the Board ID Memory says it cannot supply enough power for USB host.