How to copy ProDos hard drive volume, including invisible files?

[Jack Countryman <jcountry@seidata.com>]

We've got an old hard drive at school that runs a set of MECC software
that the kids use for drill in a lab of IIEs.  The hard drive seems to
be failing.  We'd like to move the contents of that hard drive to a
different drive so we can keep the lab going.  When I try a file copy of
all visible files, or a volume copy of that hard drive volume to the new
drive in GS.OS, I end up with 46K or so less space used, but with the
same number and size of visible files.  Several critical programs then
do not work.  So, there must be hidden file(s) of some sort involved. 
Does anyone have ideas on how to copy all of those?  If I do a
backup/restore of the hard drive volume (once I remember how) will that
transfer all those invisible files?  Or is there some way to mount the
IIGS (ProDos 8 format) hard drive to a mac desktop so I could make a
disk image that would include everything?  (So far the macs want to
format the drive at startup, then will not show it once I choose not to
format...even with PC Exchange, etc. loaded.  What else would get the
Apple II hard drive volume to show on the mac desktop?)

Ideas welcome.