If you have concatenated the chunks received & could uncompress the composite buffer, it sounds like the sub that gets triggered in the add_header is being passed a part of the same gzipped data stream every time it is invoked. You can push the compressed data a buffer at a time through Compress::Zlib. Something like this

use Compress::Zlib;

my $gunzip = inflateInit(WindowBits => 16 + MAX_WBITS)
       or die "Cannot create a inflation stream\n" ;
...
    $mech->add_handler( response_data => sub {
      my ( $response, $ua, $h, $data ) = @_;
      my ($buffer, $status) = $gunzip->inflate($data);
      # uncompressed data in $buffer

      # return true to get called again for same response.
      1; 
    } ;
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