$ff->file() is an unfortunate name, it is documented as "The name of the remote file". At least for HTTP, a document available via a URI might come from a database, or it might just be generated on the fly, so there might be no file at all.

Plus, calling $ff->file(...) before calling $ff->fetch(...) will change the source URI for at least some URI schemes and fetch methods, which is NOT the desired behaviour.

But: The documentation for $ff->file() has a second sentence, and that's the real hint:

For the local file name, the result of $ff->output_file will be used.

It might look like you could simply call $ff->output_file("document$i.pdf"), but a quick look into the source code shows that more work is needed:

sub output_file {
    my $self = shift;
    my $file = $self->file;
 
    $file =~ s/\?.*$//g;
 
    $file ||= $self->file_default;
 
    return $file;
}
[download]

So, you need to replace output_file, either by monkeypatching or by inheriting from File::Fetch and replacing that method with one that returns the desired file name.

Or, if all you need is to fetch from HTTP or HTTPS, follow hippo's++ advice and use LWP::Simple.

Alexander