I would call Pulsar NA support 1-817-225-0310.
It sounds like the unit is not charging. When you insert the battery into the Charging Adapter Plate, it will show a battery status for a few seconds. I believe Red = bad battery or bad connection, solid Green = full charge, and if blinking it tells you how full by how many blinks. However, after a few seconds the light goes off. Your description sounds normal but when you plug it in, it should continue to blink (charge) until it shows solid green, and that solid green should stay on even when done charging. To see if it is the plug, you could connect the cord up to a portable battery pack if you have one. If it works, it is the USB Plug. Another test would be to see if you connect the USB directly to the scope, and if it charges, then it is the Charging Adapter Plate (not sure what to call it). Either way, Pulsar support should be able to help you diagnose and help replace any defective items.