Maybe the problem stems from some other module loaded further up in the module hierarchy?

For debugging such things, I usually dump %INC together with the loaded module versions in the hope of finding what makes the difference:

#!perl -T

use strict;
use warnings;

use Test::More tests => 1;

my $module;

BEGIN {
   $module  = "WWW::Mechanize::PhantomJS";
   require_ok( $module );
}

diag( sprintf "Testing %s %s, Perl %s", $module, $module->VERSION, $] 
+);

for (sort grep /\.pm\z/, keys %INC) {
   s/\.pm\z//;
   s!/!::!g;
   eval { diag(join(' ', $_, $_->VERSION || '<unknown>')) };
}
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In your case of the program, maybe put the dumping part into an END block...