From what I've seen a 36V LiFePO4 should show more than 36V when fully charged, at no load, more like 40V. Since buying a battery just to see if that's the problem is expensive guess work, as is buying a controller, there may be a cheaper alternative.

If you loaded the battery with a 2 ohm resistor it should draw about 20 amps. Problem is that's 720 watts and most resistors can't handle that. But, you could buy a bunch of relatively inexpensive power resistors from Radio Shack or better yet, a local electronics store, then parallel them to build up the power rating to some respectable level. So you could but 10 20 ohm resistors and parallel them to get your 2 ohms, if each one could handle 20 watts you'd have a 200 watt dummy load.

The experiment may toast the resistors if the total power rating is less than the planned power pull, that's why you want to find cheap ones. And all the the power draw is converted to heat so I'd set the dummy load on cement or metal, not the family dining table.

Put your volt meter on the battery and touch the load, if it cuts out then you most likely have a battery/BMS problem, if not, it's more likely a controller/throttle issue.