Yes, any of the popular caching plugins should work. Just install one and try to get a sense of what it’s doing. For debugging issues, it might be useful to temporarily disable caching. That way you can confirm that it’s not a cache related issue.
Comment counts are set from the function that’s returning this error:
[2024-03-14 06:02:59] comment.ERROR: sync_comments.response_error {"message":"Forbidden","discourse_topic_id":"","wp_post_id":13514,"http_code":403}
If you are still getting that error, comment counts will not get updated.
Having the Proxy Real IP - 1.1 plugin installed on the site suggests that its server configuration is fairly complex. Maybe check with the person who setup the site to ask for details. You might also consider moving the site to a hosting service that would take care of this kind of thing for you.
I’m less certain than I was yesterday that the issue will show up in the Discourse error logs, but it’s still worth trying.