If you look at the documentation in perlop for backticks it provides several strategies; shell redirection of STDERR to STDOUT, swapping STDERR and STDOUT, and even just writing STDERR to a file that you can look at later. It's entirely possible none of these are convenient for your use case. I'd recommend instead going to a module like Capture::Tiny where your code would look something like this:

use Capture::Tiny qw(capture);

my ($stdout, $stderr, $exit_code) = capture {
    system('htmldate', '-u', $url)
};
[download]

I used the list form of system as well, to avoid exposing $url to the shell; this is a bit of a safety best practice.

You can see here that using the capture function you can run your external call as a system call, and then capture STDOUT, STDERR, and the exit code into distinct variables that you can then inspect after the call.

The Capture::Tiny module consists of less than 430 lines of well-tested code, and has no non-core Perl dependencies, so it should be quite easy to install.

The copy of the FAQs here in the monastery to which you linked (good for you for finding them, reading them and linking to the relevant one) are unfortunately well out of date.

The maintained FAQ has some better advice such as using IPC::Open3, etc.

For capturing both STDOUT and STDERR I'd suggest IPC::Run3 or Capture::Tiny. For all the details and example code see my node here.