If there are fatal errors on the site there should be details in the logs, do you have access to those?
The least disruptive thing would be if we take a copy of your site to install on a local test server, and we can update everything and see if we notice any errors.
Let me set up a private reply so that we can get credentials from you to access the site if that's okay.
You may want to create a temporary admin user for us to use that you can later delete. And be sure to have a current backup of your site.
Are there any particular actions on your site you take that provoke the error? Editing some page maybe?
Since I am overwhelmed with other duties right now, I would like to reopen this case when you have another update since that is what causing my site to go down. I would like to consult with you on what to do before I run the update.