I'd also warn you that many beech sprout from the roots of mature beech, so this little bugger may not have enough of its own roots to sustain itself if dug up. Dig a bit around the roots and see if that's the case. If so, you could try to chop the mature root on the far side of this tree (so that it is still connected to the mature tree, but the root ends at this tree). That may get you enough roots to collect the tree after a year or so in its current place.
Rockm is giving good advice as well. The only interest I see on this base is if you could find a rock that closely fits the cavity to make it appear that the tree had grown around the rock.
CW