Why would this diet, and the Atkins diet, be causing a
loss of bone density? What can be done to improve bone density, other
than lifting heavy objects? ( I know I should be doing that.)

Why would you assume the diet is causing the problem? Some people are born
with anomalies. The diet is probably helping you.
My family has a gene defect that causes the kidneys to leak calcium into the
urine...all of us. I am the only one of us  on a paleo diet in the family
and I am the one who has the lowest loss (yes still a loss...though lower
than everyone else).
You might want to have a 24 hr. urine test to see if you are loosing. The
solution is simple. Some on this list may not like it because it is a drug,
but there are lots of studies to be found on the net that support its
effectiveness. The drug is hydrochlorothiazide... it stops the loss of
calcium from the kidneys...builds bone and most importantly reduces
fractures (some drugs build bone but don't lower the risk of fractures
because they *preserve* inferior bone). In any case it's probably not, not,
not the diet.