Support

I have two action items for you.

First, update Akeeba Backup to 9.9.5. 

Your issue is fixed since 9.9.3, but you're still on 9.9.2. Here's the release notes item from https://www.akeeba.com/download/akeeba-backup/9-9-3.html relevant to your issue:

Alternate Configuration page saving method which doesn't hit maximum POST parameter count limits. Our backup profile configuration page has gone over 100 individual configurable parameters. When this is submitted back to the server, it would trigger the maximum POST parameters limit in a stark minority of servers which expect 100 or less of these parameters. We are now using client-size JavaScript code to serialise the configuration to JSON before submitting it to the server as a single, fairly small JSON document, therefore overriding the observed issue.

Also note that you normally get an HTTP 500 Internal Server Error when that happens. However, your site is configured with custom Apache error page. However, these pages point to files which do not exist. When Apache sees PHP's HTTP 500 and tries to look for the custom error file to use in the response, it can't find it. Hence the 410 Gone – depending on configuration it can also manifest itself as a 404 Not Found.

So, second action item: ask your host to disable custom error pages for your site. This will let you see the real error codes on your site, avoiding the confusion of a rare HTTP status like 410 Gone.