3,6 Volts seem to indicate the Lithium-Ion battery has only one cell, so the third wire may not be intended to balance charging.
I found some web site that sell 3 wires 3,6 Volts battery and the third ‘white’ wire is marked as a ‘thermistor wire’. (Just for my ref... what is a thermistor ? Here :
Overview | Thermistor | Adafruit Learning System
See an example here :
eBay
In an other forum, there is a poster that gave this information :
Most of two-wire batteries are made cheap for RC (Radio-Controlled) toy applications. They have super-high discharge rates (15C to 90C) and can be harmful.
For individual replacement and for your private use you can do whatever it takes, but if you aim at industrial scale, consider all warnings in Olin's answer.
I have seen the opposite, an original China-made battery would came with three-wire connection form its BMS protecting circuit, but a European-based medical supplier re-packaged it with the yellow thermistor wire cut-off, making it a two-wire connector. So go figure the safety and reliability of this re-branding.
Your device will likely need the thermistor connection (usually PTC102, aka 10k to ground) to operate, otherwise the charger chip will sense a faulty battery (like most of Texas Instruments chargers would do, for example). You can re-use the old thermistor, or just fake it with corresponding chip resistor with a nominal value (10k at 25 deg.C, or whatever it is).
NOTE: there is an informal convention on pouched batteries dimensions. This one is labelled as "6043450", which means that it is 6.0 mm thick, 34 mm wide, and 50 mm long. Just FYI, you can use this as a search parameter to find a better fit.
So there is also a possibility for the battery to be Ok, but the white wire of the thermistor sensor or the thermistor itself is faulty and the onboard charger of the SC3 does consider this battery as damaged and refuses to charge.
There may be a problem with the battery itself or the SC3 charger.
To test the battery itself, the only way to check it would be to charge the battery with an external 2 wires (black/red) charger and see if the battery charges and keeps its load.
To test the thermistor white wire, the chip resistor could be faked with a corresponding value, or exchange the exact same battery reference with a new one with 3 wires. Or more simply if you have an ‘universal electric checking device’, you can try to check the resistance between the white wire and the black one is 10kOhm when the external temp is 25°C. Testing a thermistor here :
Overview | Thermistor | Adafruit Learning System
Then, if it does not work, it may come from the SC3 charger, or there may be a real ‘overheating’ problem with the SC3 like a shortcut or a battery problem that causes the thermistor to detect a high temp and the SC3 think it should not charge the battery...
... as all this process is long and risky, and as Parrot did accept to exchange your SC3... that’s your best option !
As you are not the only one with this problem, it would be interesting to follow-up with the question...
When you mention a 0 volt indication, is it between the red and black wire of the battery ? Or from the charger ? From the white wire (that would make sense if it is a sensor) ?
You are mentioning an ‘odd circular battery’, does it look like this one :
BatterySpace.com/AA Portable Power Corp. Tel: 510-525-2328 - Powerizer Battery Official Site. Offer Lithium ion batteries, LiFePO4 batteries, polymer batteries, LiMnNi batteries, LiNiMnCo batteries, Nimh batteries, nicd batteries, lead acid batteries, primary batteries, second batteries, battery...
www.batteryspace.com
More on thermistors (1$ for a pack of 10) use in electronics here (for my reference) :
A thermistor is a thermal resistor - a resistor that changes its resistance with temperature. Technically, all resistors are thermistors - their resistance changes slightly with temperature - but the change is usually very very small and difficult to measure. Thermistors are made so that the...
learn.adafruit.com
I am no specialist in electronics or electricity, but I just try to understand the root cause of the problem...so be careful if you test things in your SC3 !