Unfortunately, this needs is still unable to be achieved.
No matter what permission settings you've made for this field before, once you've done this step:
Then I'd like a native field in the calendar to read (import) the value from the approvals list.
We still have no control over who modifies it in the calendar list.
As you know, SharePoint doesn't do user restrictions at the field level, if you're good at code, maybe you can try the methods mentioned in the following article:
Field level permission development
Please note: Microsoft provides third-party contact information to help you find technical support. This contact information may change without notice. Microsoft does not guarantee the accuracy of this third-party contact information.
In addition, I suggest you vote and monitor the issue in user voice, if a feedback is high voted there by other customers, it will be promising that Microsoft Product Team will take it into consideration when designing the next version in the future.
Support Field level Permissions
If the response is helpful, please click "Accept Answer" and upvote it.
Thank you for your understanding.