On external power (usb as a power source only, no computer), it boots and operates for perhaps 10 to 20 seconds then shuts down and then restarts and repeats the cycle. It will not turn on with internal battery. With usb from computer it goes to the black screen with the following message:
Boot version: 1.0
arg=CAFEBABE addr=40000000
USB: Connecting
Bootloader USB mode.
Battery voltage on battery measures 3.18 volts across outer terminals. There is a third (middle terminal) battery wire. I am not sure of its purpose. It measures 2.91 volts to one of the terminals and 0 to the other.
I am not sure, what the problem is. But, you could try to boot to OF and charge the battery.
My feeling is, that Fuze+ in Rockbox mode is not able to charge battery, if battery is
deeply discharged. At least
my Fuze+ is not able.
Once I had the same behaviour. The player booted (while USB charger plugged in), showed B&W boot screen and rebooted again in a few seconds. I guess the reason for this was, that battery capacity was so low that, it rebooted before it reached the code for charging... I thought, player is dead. But switching to OF and recharging helped. It still lives, more than year later. :-)
Switching to OF is done by holding On together with Vol+ (or Vol- .... I am not sure which one of those two)
Also,
your bootloader is old. I am not sure, but I think that the newer one has also some improvements for battery charging.