On the plus side, it's possible, and you don't even need overloaded operators.

use strict;
use warnings;

package MyStruct;

sub new {
   my $class = shift(@_);
   return bless({@_}, $class);
}

sub Counter : lvalue {
   my $self = shift(@_);
   $self->{'Counter'} = $_[0] if (scalar(@_));
   $self->{'Counter'}
}

package main;

{
   my $x = MyStruct->new(Counter=>0);  print($x->Counter, "\n");  # 0
   $x->Counter($x->Counter + 1);       print($x->Counter, "\n");  # 1
   $x->Counter = $x->Counter + 1;      print($x->Counter, "\n");  # 2
   ++($x->Counter);                    print($x->Counter, "\n");  # 3
   ++$x->Counter;                      print($x->Counter, "\n");  # 4
   $x->Counter++;                      print($x->Counter, "\n");  # 5
   $x->Counter += 1;                   print($x->Counter, "\n");  # 6
}
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On the downside, you can't use Class::Struct with this method, or you have modify it slightly. I think the changes are limited to the following:

Change (untested)
$out .= "  sub $name {$cmt\n    my \$r = shift;\n";
to
$out .= "  sub $name : lvalue {$cmt\n    my \$r = shift;\n";
and scalar accessors will be returned by lvalue. Array accessors and hash accessors need a little more changes, because they use 'return' than that wipes the lvalue-ness.