A lock ring uses a left hand thread AND is smaller in diameter than the threads on a free wheel or a track cog. therefore, the lock ring will not fit on the freewheel side of a fixed free flip flop hub.
A free wheel tool is relatively cheap. But if you really want to go without one, what you need to do is to figure out a way of grabbing the "dogs" on the "stationary" part of the freewheel and hold that steady. Then you turn the wheel by the rim or tire and that will unscrew the freewheel. (this is for the usual shimano type ss free wheels) good luck