Not only is there nothing "special" about my configuration, but there's no
longer anything wrong!  On a hunch, I looked in my wastebasket for the
printouts where there was no text transfer, and I found that the Bar Code was
populated properly!  

So I started from scratch and put in the bar code line, leaving it where
Mail Merge Helper insisted -- at the top.  I ran, and all went well.  

Then I edited the envelopes document, cutting the bar code line from the top
and pasting it at the bottom where I really wanted it, and again all went
well.  (This surprises me, in view of the troubles I describe next.)

Then I re-created the envelopes document, inserting the merge fields first.  
I positioned the cursor two lines below the bottom address line, and tried to
insert the bar code line, but it insisted on going to the top.  I tried to
cut/paste it below the address, but it fought back and wouldn't allow it.  
Finally, I re-created it, inserting the bar code line first, then I inserted
a couple of carriage-returns at its start, then inserted the other merge
fields above it.  That worked.

We still don't know what went wrong originally.   I got it working, and
maybe the techniques I used (good and bad) will be of value to someone else.

Thanks again for all your time.  It's been fun -- sort of.  :-)

Bud