Interesting. I was having a similar problem and I solved it by making each of the handlers closures with a shared lexical. While I had each handler as a named subroutine it is also probably better written as a generator. I'll include samples of each mode - what I actually used and what I think is better.

# My actual implementation
{
    my $object;
    sub setobject {
        $object = shift;
    }

    # Each of these routines has access to the shared
    # lexical. Enclosing this whole section in a block
    # prevents the lexical from being used in other scopes
    sub start_handler { ... }
    sub char_handler { ... }
    sub end_handler { ... }
}

# What is probably better though there is a heck of a lot
# of space to move this around. I implemented this as a
# wrapper over some named subroutines and passed the shared
# variable in. You can really do this however you feel.
sub get_handlers {
    my $object;
    return {
      start_handler => sub { start_handler(\@_,\$object) },
      end_handler => sub { end_handler(\@_,\$object) },
      char_handler => sub { char_handler(\@_,\$object) }
    }
}
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