You can do a lot of DistoX2 splays and then try to manually match the LIDAR point cloud to the DistoX2 splay model. This is the method I used twelve years ago when I was first experimenting - it is very time consuming. Now I would suggest you use something that is sufficiently large enough to register in the LIDAR cloud that can be surveyed with the DistoX2 eg. a large distinctive rock, formation or other natural feature or something man-made, tackle bag, container, helmet etc.
I believe this purpose could very easily be served by the "three white orbs". (Though the accuracy might not be great, since they are not points, so you are hitting somewhere on a surface, and can't precisely locate that.) But this also then brings out the potential for error when the post processing doesn't match the centreline, and one of them has to "win". As you said, this is done in mine surveys. I still wouldn't want to be using Therion to draw up LiDAR data, just because there is so much that it becomes unusable. It is already hard enough drawing up a survey that has a lot of regular splays because the team got over-enthusiastic with the Disto. "Wait, which one of those splays points to the pitch edge?". And there is nothing to tell you if something is a pitch, a climb, a floor step, or a dangerous pile of loose boulders. Point clouds are great for showing fancy fly-throughs for non-cavers, and for mining engineers to see exactly how much more material they can remove before compromising a wall, but they add a lot of confusion for a cartographer. (But again, see Julian's project.)