This is the continuation of a discussion started here: http://www.swiftideas.com/forums/topic/gravity-forms-product-add-ons/
In short, the products that use GFPA do not add to cart if Ajax is enabled, but they add properly if Ajax is disabled. The products that do not use GFPA do not add to cart when Ajax is disabled, but they add properly if Ajax is enabled.
I don’t use a caching plugin, but I use WP Engine, who handles caching internally. I contacted WP Engine and they originally guessed that it was likely a caching issue, so they added the rules they recommend for sites that use WooCommerce. After doing that, the issue persisted and they said that it does not appear to be a caching issue, which seems to make sense because the issue resolves when I switch to a different theme.
For clarity, the product category that uses GFPA is Shop –> Custom Fabric Bulletin Boards. None of the other three product categories under Shop use the GFPA extension.