Look, Akeeba Backup only has one plugin which may interfere with third party extension installation: "System - System Restore Points". When you disable that plugin absolutely zero lines of Akeeba Backup's code run when you install a third party extension. Code which doesn't run doesn't affect a system. Please triple check that the plugin is question is disabled.
Side note: there is also another plugin which may interfere with your site's operation, the "Akeeba Backup Lazy Scheduling" one. This plugin is deprecated since several versions ago and we strongly advise not using it. If you do use it please disable it.
When accessing the Extensions, Manage Extensions you must not see the Akeeba Backup logo at the bottom of the page, followed by the words "What is this • Back to normal installer". If you do see that line the "System - System Restore Points" plugin is not disabled and you must disabled it – sometimes you may have to log out, close your browser, open it again and log back into your site due to some browsers stupidly caching redirects (quite complicated to explain, just trust me on that). If you do not see those words and you still have this issue and still don't trust that I know what I'm talking about there is always the nuclear option: uninstall Akeeba Backup. This won't fix the issue, but it will make it clear that it's not related.
Finally please note that the issue you are having indicates a timeout issue. Your server is too slow and the extension package is too big. This causes PHP to take a very long time processing the upload, extraction, SQL script execution and file copy which is required to install an extension. If you go over the max_execution_time value set in your server's php.ini file you get a timeout error which usually manifests itself on production hosts either as a blank page or an internal server (500) error page. This kind of issue is NOT specific to Joomla! and is NOT caused by Akeeba Backup. It's just how web servers and PHP work. And your host cramming too many sites on a single server, of course. If you try at a different time in the day or a different day the server might be less busy and not see that error.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!