Your example is quite unclear to me. What is this supposed to to?

sub mylol {
  my $self=shift;
  {
    local $self; $self->{data} = {newdata => 1};
    $lol = super();
  }
}
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You could simply localize ->{data} and be on your merry way:

sub mylol {
    my $self = shift;
    local $self->{data} = { newdata => 1 };
    super();
    # $self->{data} is restored here
};
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(or, depending on what super() does, just ditch whatever contraptions Moose creates, and directly call $self->SUPPER::mylol, to achieve your goal).

Update: After having looked at Moose::super, I'm not really sure what your aversion is against package variables, as it uses them as well. And your locally setting $self is quite unlikely to have any sane (or desired) effect as likely the referenced global variables in Moose:: get set from a wrapper installed by Moose::override:

package Moose;
...
our $SUPER_PACKAGE;
our $SUPER_BODY;
our @SUPER_ARGS;

sub super {
    # This check avoids a recursion loop - see
    # t/bugs/super_recursion.t
    return if defined $SUPER_PACKAGE && $SUPER_PACKAGE ne caller();
    return unless $SUPER_BODY; $SUPER_BODY->(@SUPER_ARGS);
}
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I'm not sure what that contraption is supposed to achieve, but then, maybe that's because I have either never encountered the problems that Moose attempts to solve or have always taken other approaches to these problems.