Ok lets look at what gives the motor power. You took the controller out of the problem by uses a good one.

The 36V 10Ah Lithium battery has a BMS. If the BMS is overheating then it will cut power to the controller. Also if the motor is pulling to many amps then the BMS will cut power to the controller. By pulling the fuse and resetting the system it resets the BMS as well. Thats why I think it's the BMS.

As far as the battery goes. You may have a bad cell in the pack. This will show up by a lower voltage at full charge with a volt metter. Li-Po cells come 3.6v per cell. So for a 36v pack you should have 10 cells at 3.6v per cell. Now if you have a bad cell it will show only 32.4v. Now the volatge cut off for a 36v battery is about 29v. Thats 80% of the full charge of the pack. So if you started out with a 32v volt charge it will cut out a lot faster then it is rated for. Because the BMS is doing it's job.

in short it's the BMS or the battery. Hope this helps