No, unfortunately you're not able to include a free gift product when a customer checks out with shopify's accelerated checkout buttons (Shop Pay, Apple Pay, Google Pay, ect) directly from the product detail page (PDP) of your site.
This is because when clicking the Shop Pay or Apple Pay button it bypasses the cart and takes the customer straight to checkout.
Shopify doesn’t provide any way for apps to hook into the action in order to add the gift product to cart before the redirection to checkout occurs.
Shopify also doesn’t allow apps to add products to the cart on the checkout page.
So unfortunately this is a Shopify limitation and not an app limitation.
You can usually disable shopify's accelerated checkout buttons on product detail pages in your theme settings. This will ensure customers use the regular add to cart button and that the GWP is added to the cart. If you are unable to do this in your theme please reach our to your theme developers for assistance.