As mech-dump indicates that the button you are clicking is the submit button, instead of clicking the button I would use the submit function from WWW::Mechanize, as follows :

...
$mech->untick(SSview => "on");
$mech->submit;
       
die $mech->res->status_line unless $mech->success ;

my $uri = $mech->response->request->uri->as_string ;
print $uri ;
[download]

Is it better ?

We discussed form submitting in how to submit html form?.

Hope this helps.

Gu