Is there a simpler solution?

DIY :)

Is there a different drop-down list widget that is displayed as only a button?

Don't know , maybe, would investigate tk j ... Tk::JFileDialog Tk::JComboBox Tk::JBrowseEntry ...

Is there a simpler solution?

Looking at a widgetdump, you can do  $jb1 ->Subwidget('entry')->packForget ; that works to hide it

this messes up the width, so you'd have to bind the button to pack before invoking original button handler

{
    my $parent   = $jb1 ;
    my $arrow    = $jb1 ->Subwidget('arrow');
    my $callback = $arrow->bind('<Button-1>');
    $jb1 ->Subwidget('entry')->packForget;
    $arrow->bind('<Button-1>', [
        sub {
            my( $button, $jbl, $callbackarray ) = @_;
            warn "fucking @_\n";
            warn "button($button) jbl($jbl) callback($callback)\n";
            $jbl->Subwidget('entry')->pack(-side => "right", -fill => 
+'x', -padx => 0, -pady => 0, -expand => 1); 
            my( $cb, @args ) = @$callbackarray;
            $cb->(@args);
        },
        $parent, $callback,
    ] );
}
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This is only half ... if anything it tells you what to edit in Tk::JBrowseEntry :D in Tk::BrowseEntry::SetBindings ...

update:

#!/usr/bin/perl --
use strict; use warnings;
use Tk;
use Tk::BrowseEntry;

Main( @ARGV );
exit( 0 );

sub Tk::BrowseEntry::Popdown {
    my ($w) = @_;
    if ( $w->{'_BE_savefocus'} && Tk::Exists( $w->{'_BE_savefocus'} ) 
+) {
        $w->{'_BE_savefocus'}->focus;
        delete $w->{'_BE_savefocus'};
    }
    if ( $w->{'_BE_popped'} ) {
        my $c = $w->Subwidget('choices');
        $c->withdraw;
        $w->grabRelease;
        if ( ref $w->{'_BE_grabinfo'} eq 'CODE' ) {
            $w->{'_BE_grabinfo'}->();
            delete $w->{'_BE_grabinfo'};
        }
        $w->{'_BE_popped'} = 0;
        $w->Subwidget('entry')->packForget;
    }
}
sub Tk::BrowseEntry::BtnDown {
    my ($w) = @_;
    return if $w->cget( '-state' ) eq 'disabled';

    if ($w->{'_BE_popped'}) {
        $w->Popdown;
        $w->{'_BE_buttonHack'} = 0;
    } else {
        $w->Subwidget('entry')->pack(-side => 'right', -fill => 'x', -
+expand => 1);
        $w->update;
        $w->PopupChoices;
        $w->{'_BE_buttonHack'} = 1;
    }
}


sub Main {
    my $mw = tkinit();
    my $var = 'lelabel';
    my $b = $mw->BrowseEntry(-label => "Label", -variable => \$var);
    $b->insert("end", "opt1");
    $b->insert("end", "opt2");
    $b->insert("end", "opt3");
    $b->Subwidget('entry')->packForget;
    $b->pack;
    MainLoop;
}
[download]