I made the following changes:

• call to $one->new sends \$dbh instead of $dbh.
• One::new expects \$dbh instead of $dbh, and dereferences it.

That's it. I didn't expect it to work, and only tried it because the symptom seemed to be that of a variable passed by value, not by reference. The fact that it did work has left me bewildered, and is the reason I started this thread.