These customization fields are used if you are simple user and you are not designing a module. Using it to save the designID will be just a hack.
Since you're creating a new module my advice is to keep the 3rd party ID in a newly created database table, which will match the id_product, id_design, id_cart, id_order, etc...
You can hook to "actionCartSave" and add the record with the matching ids, and all the other required data at your table.
If you want to stick to that Customization feature, add a sample field and review the following database tables:
ps_customization ps_customization_field ps_customization_field_lang ps_customized_data
and replicate the changes when you receive the 3rd party ID.
If all your products will be customized, consider adding the required data in ps_customization_field & ps_customization_field_lang (the table for the field structure) during the module installation, so after it's installed you can just fill ps_customization & ps_customized_data (the tables for the field data)
PrestaShop does not have proper API for adding customizations, only for retrieving data, so you'll have to write the SQL queries yourself. Just review the ProductController for the ps_customization & ps_customized_data changes and the AdminProductsController for ps_customization_field & ps_customization_field_lang.
Do not forget to remove the Customization markup code from your product & cart templates.