How do I safely force get_theme_mod() to use a defined get_option(“stylesheet”) value?

Filter "theme_mod_$name" in a separate plugin:

add_filter( 'theme_mod_very_special', function() {
    return 'forced value';
});

To replace the complete theme mods, filter pre_get_option_theme_mods_$theme_name and return an array. You have to register that filter for each theme name.