Thanks.
I tried your code posted as an update, and it mostly does what I want, but the entry field still shows up when I click the button. I would want it to stay hidden all the time. I tried removing the whole BtnDown sub, removing just the $w->Subwidget('entry')->pack line, and adding a packForget right after it. The result is the same in all three cases: the entry field is hidden, but the listbox that pops up starts at the left edge of the screen and stretches all the way to the button. I tried looking in the BrowseEntry.pm to see what sets the width but most of it goes way over my head. Adding -listwidth   => 30 brings me back to where I started: the listbox is the correct width but it's at the edge of the screen.