You won’t get any joy with customer support. Rechargeable batteries, of any sort, are not supported.
From their own FAQ:
"Blink cameras are powered by two non-rechargeable AA 1.5v Lithium batteries. All cameras will ship with batteries, and they’re easy to replace when the time comes. Simply slide open the battery compartment, take out the old batteries, and insert the new set.
Blink camera batteries should be replaced by Energizer Ultimate 1.5v Lithium AA batteries or the equivalent. These batteries can be purchased here.
Be sure to avoid AA Lithium Ion or Li-ion batteries as these will carry a higher voltage and may cause issues with your camera. You also want to avoid Alkaline based and rechargeable batteries, as they may cause issues with your camera.
A new set of non-rechargeable AA 1.5v lithium batteries should last for a total of 40,000 seconds of motion-activated video and/or live video. We define standard or typical use as approximately 10, 5-second video events per day."