To make it clear, I will try to explain further the difference between slugs and custom permalinks.
The main idea behind Permalink Manager is to replace the original permalinks (the slugs are a part of them) with the custom permalinks. As they can be fully customized, they do not need to have the slugs included. However, by default, they follow the original permalink structure, which includes the slugs.
The snippet I sent you will automatically update only the slugs whenever a product title is changed. This would also affect the original product permalinks (used when the plugin is disabled). You can see this clearly when Permalink Manager is temporarily deactivated.
As the plugin overwrites the original product permalinks with the custom permalinks, the custom permalinks are updated in a different (simpler) way. In the plugin settings you can enable auto-update mode:
https://permalinkmanager.pro/docs/basics/automatically-update-the-permalinks/#2-auto-update-custom-permalinks
To sum up, if you use the above snippet and have auto-update switched on for custom permalinks, any changes to the product’s title automatically update both the slug itself and slug inside the custom permalink to the same value.