I do like your idea....
I think the key is to do it at the right time and with the right technique.

I saw a tool made especially to cut out a single eye. That'd be great during the growing season.

In the spring, I think it's easiest to do when the pips are just a few inches tall. Then you can clearly see where the plant may naturally want to be divided.

For me, however, since I'm at that "critical mass" point, unless I were to start taking out half of a plant, I don't think it'd be possible for me to free up enough space. I don't have the space to start dividing and putting a dupe elsewhere in the garden.. Simply put, there's no room at the inn! Until/unless I can get someone to remove tons of rock that the previous owners put down as mulch, I'm stuck!

If the rock is loose on top , maybe someone that needs rock would rake it up? Or as far as getting rid of the rock yourself I know a guy that always references the EATING of an Elephant when talking about big jobs...

How do you eat an elephant?
One bite at a time !

Oh, I regularly tout that particular piece of wisdom.
However, until I actually NEED to remove that rock, I am NOT going to break my back over it.

It's slowly been leaving areas of my yard. The front yard is almost completely eradicated. Parts of the back yard are next on the chopping block.