WWW::Mechanize is wrong to inherit from LWP::UserAgent. It should encapsulate it. That means you need to resort to the following hack:

use LWP::RobotUA   qw( );
use WWW::Mechanize qw( );

BEGIN {
   package WWW::Mechanize;
   our @ISA;
   @ISA = (
       'LWP::RobotUA',
       ( grep $_ ne 'LWP::UserAgent' && $_ ne 'LWP::RobotUA', @ISA ),
   );
}
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Note that this will affect all WWW::Mechanize objects in your process.