Google Merchant Center increases store visibility by showcasing products across various Google platforms, complete with shipping costs. However, inconsistencies between actual shipping rates from Better Shipping and those displayed on Google Merchant may lead to Google not showing your shipping rates.
To prevent abandoned purchases, Google aims to avoid incorrect pricing data. As a merchant, you must ensure your Shopify store's shipping rates are equal to or lower than those on Google Merchant Center. For more information about this please see the How It Works section in the Google Shipping Cost Help page
Consider this example: Google displays a product with a $10.00 shipping fee, but at checkout on your site, the customer faces a $30.00 shipping charge. In this case, the customer may abandon the purchase.
To fix this issue, set the Google Merchant shipping settings higher than those in Better Shipping and and any other shipping rates you have in Shopify. Consult Merchant Center Customer Service or your Google representative for help in increasing shipping fees on the Merchant Center above your Shopify store rates.
For more information on configuring shipping options on Google Merchant Center, follow this link.