G'day ImJustAFriend,

Instead of reading one line at a time, you could read one chunk at a time by setting $/ (see perlvar: $/).

Here's a quick-and-dirty example script to demonstrate the technique. I pared down your posted data substantially; duplicated it; then labelled with "CHUNK 1" and "CHUNK 2".

#!/usr/bin/env perl

use strict;
use warnings;

{
    local $/ = "\nMessage Id";

    while (<DATA>) {
        chomp;
        /^message=(.*)\z/ms && print "$1\n";
    }
}

__DATA__
<13>Nov 15 21:26:00 OamCOMM[365626]:
TIMESTAMP=Thu Nov 15 21:26:00 2018
...
Severity=Inform
message={username:xxxxxxx...
 causeDISTINGUISH_NAME=...
... CHUNK 1 ...
label=x
}
Message Id=xxxxxxxx
<13>Nov 15 21:26:00 OamCOMM[365626]:
TIMESTAMP=Thu Nov 15 21:26:00 2018
...
Severity=Inform
message={username:xxxxxxx...
 causeDISTINGUISH_NAME=...
... CHUNK 2 ...
label=x
}
Message Id=xxxxxxxx
[download]

Output:

{username:xxxxxxx...
 causeDISTINGUISH_NAME=...
... CHUNK 1 ...
label=x
}
{username:xxxxxxx...
 causeDISTINGUISH_NAME=...
... CHUNK 2 ...
label=x
}
[download]