I think it may be a thermistor to control charging. It changes resistance to reduce or increase charging current based on the temperature of the battery pack.
The ohm reading you are getting is because of the battery voltage and flipping meter leads is changing the polarity. Resistance readings can't be done with voltage present.
A fairly simple description where the poster thought it was a diode -
https://electronics.stackexchange.com/questions/163354/purpose-of-this-diode-on-this-circuit
A deeper explanation -
https://batteryuniversity.com/learn/article/charging_nickel_based_batteries
It's hard to say if the thermistor is bad or if one of the cells in the pack is bad causing the fault light. The pack has to be torn apart to test the individual cells and it isn't simple with your pack because of the spot welded connections.
Probably time for a new battery, Jake. One thing you can try is freezing the battery before charging. I have one like that and it's on the way out cuz it isn't holding charge but it says it's fully charging when plugging the cold battery in the charger. At room temp or warmer, it shows faulty after doing the initial test before starting the charge cycle.
Can any of you electronics guys tell me what this gizmo is? I get .333 ohms through it one way and .713 through it the other way.
