And if you are planning to use that in more than one or two places, I'd monkey patch it so it's easier to maintain, and maybe submit a patch to the author–

use REST::Client;
sub REST::Client::response { +shift->{_res} }

my $uri = "...";
my $client = REST::Client->new();
# ...some setup...
print $client->POST($uri)
    ->response
    ->request
    ->as_string;
[download]

Update, shortened call to what is available by removing the ->request

Update to update, lengthened call to what was there before because it was right. :P