Interesting. I ran into a permissions problem on Kibana 7.5.1 with that same babel_register_cache.json file, but with trying to use the keystore. If you're using the keystore, that may be the problem. Go ahead and check out the topic I created to see if you think it's the same problem. Kibana 7.5.1 keystore permissions topic
To get kibana to run, I removed everything from the keystore & put it directly in the kibana.yml file. It all worked for me then. Note that I am working from a fresh install so there are differences.
My permissions on the .babel_register_cache.json file are the same as yours:
sudo ls -l /usr/share/kibana/optimize/.babel_register_cache.json
-rw-rw-r-- 1 kibana kibana 2 Jan 14 15:43 /usr/share/kibana/optimize/.babel_register_cache.json
One more thing, I have logging enabled in kibana.yml. It logs to /var/log/kibana/kibana.yml (I had to make the kibana dir and set permissions to "kibana:kibana"). That's where I got the detailed log explaining the fatal error was that the keys in the keystore couldn't be accessed. If you have logs going to /var/log/kibana/kibana.log you may get more information there. If you don't & you enable it, be warned that there is no log management for kibana so you'll need to use a service like logrotate to manage the logs or they'll just grow. In addition, there's very little documentation on how to configure logrotate for kibana. This site is helpful: Kibana - setup log rotation