I would really like a solution that just uses pack without having to use any additional libraries (JSON::XS, Encode, etc.).

Why? This very much smells like an X/Y problem to me. JSON::PP has been in the core for over 11 years (since Perl 5.14).

use warnings;
use strict;
use JSON::PP;
use Data::Dump;

my $enc = '"Alice \ud83d\ude06 Bob"';

my $json = JSON::PP->new->allow_nonref;

dd $json->decode($enc);  # "Alice \x{1F606} Bob"
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Minor edit for clarification.