Seemingly can't connect to endpointBase from AWS Server

Your elasticsearch.yml looks good. These are the only config settings in there that Enterprise Search cares about:

action.auto_create_index: ".ent-search-*-logs-*,-.ent-search-*,-test-.ent-search-*,+*"
xpack.security.enabled: true
xpack.security.authc.api_key.enabled: true

It looks like you have 2 of the 3. Maybe try adding action.auto_create_index just in case. Oh, and can you note what version you're on? I'm assuming latest / 7.9.2?

enterprise-search.yml looks like it's missing elasticsearch.host - can you confirm you have it and that it's set to the correct URL? You don't have to paste it here (obviously, for security reasons - also I'd suggest replacing your elasticsearch.password with a '' as well!), just verbally confirm that it's there.

Also a quick reminder - don't forget to hard stop & restart both Elasticsearch and Enterprise Search whenever you change a config yml file! They don't restart automatically and the changes won't take if you don't manually restart.