SkeletonRenderer/SkeletonAnimation's Render method is actually the LateUpdate method (wasn't my decision :x ).
You can call that.
Changes to skeleton are abstract and are done in in animations by Spine.AnimationState in SkeletonAnimation.Update, or by you, whenever.
Then the skeleton (and those changes) are translated into a UnityEngine.Mesh in LateUpdate.
UpdateCache repopulates the list of Updatables (bones, constraints, etc...) in case you changed something.
UpdateWorldTransform calculates world values of bones from their local values.
So it would be.
var skeletonAnimation = GetComponent<SkeletonAnimation>();
var skeleton = skeletonAnimation.skeleton;
//... do whatever
skeletonAnimation.LateUpdate();