How to secure the fork when there is no stem
Yes, this "lockring" is important. It functions like the stem on a regular bike (with a threadless fork) to hold the headset together. The cap on top is used during adjustment, but is not intended to hold everything together. See this description from Park for more details of headset adjustment: https://www.parktool.com/blog/repair-help/threadless-headset-service.

With regards to finding alternative lockrings, as Carel pointed out in the comments, there are seat clamps with the proper internal diameter, and it appears that is what your headset has instead of a stem. I can't tell from the picture for sure what your diameter is, but it is presumably either 1" (25.4 mm) or 1 1/8" (28.6mm). Seat clamps are available in both sizes. Just search for "seatpost clamp 28.6 mm". See if you can find one that is not as tall as what you have now. 

One final detail to note: In order to keep the headset together properly, the clamp must extend to just above the top of the steering tube, so if you are already very close to that height, then it is the steering tube not the clamp that is determining the height. To reduce the overall height once you have a shorter clamp, you will need to cut off a small bit of the steering tube so that it is below the top of the clamp.