I'm not sure running these tests as part of regrtest is a good idea.
Fixing and/or marking the doctests is a good idea, and having a `make doctest` that reports no error is a desirable goal.  When `make doctest` works we could think about adding a regrtest option to run the doctests too, and if these tests don't take too much extra time to run we might even consider to run them by default (or at least when -uall is specified).
Having an option to run individual doctests would be nice too, and would encourage developers to test their doctests.
You should also see if there are more doctest-safe or more doctest-unsafe tests, and mark the ones in the smallest group.