[Backport to release/11.0] Update external product button label handling - #67558

Testing Guidelines

Hi @LiamSarsfield ,

Apart from reviewing the code changes, please make sure to review the testing instructions (Guide) and verify that relevant tests (E2E, Unit, Integration, etc.) have been added or updated as needed.

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• 🖼️ Screenshots or screen recordings.
• 📝 List of functionality tested / steps followed.
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⚠️ API Documentation Reminder

Hi @woocommercebot! Your PR contains REST API changes. Please consider updating the REST API documentation if your changes affect the public API.

Changed API files:

plugins/woocommerce/includes/rest-api/Controllers/Version1/class-wc-rest-products-v1-controller.php
plugins/woocommerce/includes/rest-api/Controllers/Version2/class-wc-rest-products-v2-controller.php
plugins/woocommerce/includes/rest-api/Controllers/Version3/class-wc-rest-products-controller.php