Two ways I can think of off hand --
Working with the xml file might be the easiest way. Bring it into AME or Pr Pro and choose the in/out points. Then you can export as a looping animated gif.
If you want something other than a gif, you could export the selected in/out region as the video format of your choice, and then use one of the instant player apps with loop enabled.
I managed to do it through After Effects. Apparently the first few frames of the walk cycle in Character Animator is not part of the looping walk cycle, so had to cut them out.
I would be useful to be able to select a part of the timeline directly in CA, have it loop on playback and then be able to export only that segment.
Was your walk cycle controlled via Left/Right Arrow keys (which does have a ramp up/down phase) or set to start Immediately (which I would think would loop from the start)?
In/out range for export is a feature request we've heard before: https://character-animator.ideas.aha.io/ideas/CH-I-160