We've tested this in our development environment and couldn't replicate the issue, even with all the plugins active.
In the WPML Troubleshooting section, it states that the REST API is disabled, even though it's clearly working. We've also deactivated all other plugins that might interfere with the REST API, but WPML still reports it as disabled.
Could this be the reason for the issue we're experiencing? Is there anything within WPML itself that might be interfering with its own REST API functionality? We’d appreciate any guidance you can provide.
I don't think that's the root cause of the issue. WPML uses the REST API mainly to handle translation with the Advanced Translation Editor, and you are using the WP editor. It may be a false positive.
As you said, it's working in the Staging environment, it may be a server issue. It also worked on my environment. I suggest looking for differences between environments.