One way:

Create a subclass of LWP::UserAgent that overrides request to remove the jsessionid from the requested url.

use strict;
use warnings;

package My::UserAgent;

BEGIN { our @ISA = 'LWP::UserAgent'; }

use LWP::UserAgent  qw( );
use URI             qw( );
use URI::QueryParam qw( );

sub request
{
    my $self    = shift;
    my $request = shift;

    my $uri = URI->new( $request->uri() );
    $uri->query_param_delete( 'jsessionid' );
    $request->uri( $uri );

    return $self->SUPER::request( $request, @_ );
}

1;
[download]

Beware that this might cause infinite loops if the server insists on you using jsessionid. Well, not quite infinite since LWP::UserAgent stops redirecting after max_redirect redirects (default is 7).

Update: Fixed bug mentioned in reply. Added use strict; and use warnings;.