Unfortunately apps (including Product Customizer) cannot be used with the Shopify "buy" button (formerly the Shopify Widget) on external websites. The Shopify Buy button only includes data for your products that's been added in Shopify, and will not include data from other apps such as Product Customizer.
We'd love to be able to work around this, but we're not able to as the Buy button is generated by Shopify itself to work externally. It does not include any options aside from the built-in variants as a result. This would only be possible in the future if Shopify added support for passing back line item properties in the purchase (which is what our app uses) within the Buy button itself.