my $where = $ff->fetch( to => '$path');
Again, you are enclosing a variable inside single quotes, so it won't be interpolated. Remove the quotes for more success.
Here is an SSCCE showing how to use File::Fetch successfully.
#!/usr/bin/env perl
use strict;
use warnings;
use File::Fetch;
use Test::More tests => 1;
my $url = 'https://www.perlmonks.org/?part=1;displaytype=displaycode;n
+ode_id=11145569;abspart=1';
my $ff = File::Fetch->new (uri => $url);
my $loc = $ff->fetch (to => '/tmp/');
ok -f $loc, "Downloaded file to $loc";
Try examining the return value of fetch like this in your code to find the saved file.