To add stable sorting for the last two categories, restrict the sort to $type and $ord (packed as NN so taking 8 bytes) unless the $type is 0 for both terms, in which case sort the entire strings.

use strict;
use warnings;

use 5.014;

my @strings = qw{
   cabin
   prawn
   eminent
   pail
   error
   reptile
   engine
   copy
   echo
   soap
   eskimo
   carpet
   elias
   emblem
   skunk
   cheap
   };

say for
   map  { substr $_, 8 }
   sort {
          unpack( q{N}, $a ) || unpack( q{N}, $b )
          ? substr( $a, 0, 8 ) cmp substr( $b, 0, 8 )
          : $a                 cmp $b;
        }
   map  {
          my( $type, $ord );
          if ( m{r} )
          {
              $type = 0;
              $ord  = 0;
          }
          elsif ( m{^e} )
          {
              $type = 1;
              $ord  = ord substr $_, 2, 1;
          }
          else
          {
              $type = 2;
              $ord  = 255 - ord substr $_, -1, 1;
          }
          pack q{NNa*}, $type, $ord, $_;
        }
   @strings;
[download]

The output, note that 'eminent' and 'elias' have the same 3rd character but now retain their relative order, as do 'soap' and 'cheap' with matching final characters in the last category.

carpet
error
prawn
reptile
emblem
engine
echo
eminent
elias
eskimo
copy
soap
cheap
cabin
pail
skunk
[download]

I hope this is of interest.