Here's the solution I finally ended up with. First, I build the command string using the parameters hashref:

my $cmd = "$baseURL/$command?" .
  join( '&', map { "$_=" . uri_escape( $parameters->{$_} ) } 
    keys %$parameters );
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Then I build the HTTP::Request object, adding the Host parameter (as a flattened hashref), and set the request's protocol:

my $request = HTTP::Request->new( 'GET', $cmd, [ 'Host', $host ] );
$request->protocol('HTTP/1.1');
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Finally, I use LWP::UserAgent to send the request:

my $agent = LWP::UserAgent->new();
my $response = $agent->request($request);
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So it looks like both the HTTP/1.1 at the end of the request and the Host parameter in the header were required.