Update: I think I'm closer:
RFC-4180, paragraph "If double-quotes are used to enclose fields, then a double-quote appearing inside a field must be escaped by preceding it with another double quote."

So:

2,""Rat Control" <sip:+15559999999@192.168 .5.233>;tag=gK004bb052",9
[download]

is malformed, incorrect CSV, this should be:

2,"""Rat Control"" <sip:+15559999999@192.168 .5.233>;tag=gK004bb052",9
[download]

I made some experiments. Here are my (updated) results:

#!/usr/bin/perl
use strict;
use warnings;
$|=1;  ## turn off buffering for STDOUT
 
use Text::CSV_XS qw( csv );
my $csv = Text::CSV_XS->new();  #using the defaults

while (my $line = <DATA>) 
{
  if ($csv->parse($line)) {
      my @fields = $csv->fields();
      print join ("|",@fields),"\n";
   
  } else {
      warn "Line could not be parsed: $line\n";
  }
}
=Prints:
1|Rat Control <sip:+15559999999@192.168 .5.233>;tag=gK004bb052|9
2|"Rat Control" <sip:+15559999999@192.168 .5.233>;tag=gK004bb052|9
3|Rat Control <sip:+15559999999@192.168 .5.233>;tag=gK004bb052|9
Line could not be parsed: 4,"Rat Control" <sip:+15559999999@192.168 .5
+.233>;tag=gK004bb052,9

5|123,456|abc
6|Rat|xyz
7|Rat Control|xyz
Line could not be parsed: 8,""Rat Control" <sip:+15559999999@192.168 .
+5.233>;tag=gK004bb052",9

=cut

__DATA__
1,Rat Control <sip:+15559999999@192.168 .5.233>;tag=gK004bb052,9
2,"""Rat Control"" <sip:+15559999999@192.168 .5.233>;tag=gK004bb052",9
3,Rat Control <sip:+15559999999@192.168 .5.233>;tag=gK004bb052,9
4,"Rat Control" <sip:+15559999999@192.168 .5.233>;tag=gK004bb052,9
5,"123,456",abc
6,"Rat",xyz
7,"Rat Control",xyz
8,""Rat Control" <sip:+15559999999@192.168 .5.233>;tag=gK004bb052",9
[download]

I do not understand why Line 4 which starts with unnecessary quotes is not parsed? update: But could be that the double quotes must apply to the whole field and therefore the syntax in line 2 must be used.See Line 5 which has an embedded comma and requires the quotes and is parsed correctly. See Lines 6, 7. I don't think the starting quotes are the issue, it appears that other "special" characters in Field2 are causing the problem.