If you have followed the eighteen step process I linked to in 35062 AND you have re-published the installer plugins but you are still getting a 403 it means that you are entering the wrong Download ID in Akeeba Backup.
Log into our site and click on Add-on Download IDs in the menu right under our logo. It takes you to https://www.akeeba.com/download/official/add-on-dlid.html You will see your Download ID there. Follow the eighteen step process once again using the new Download ID.
If you still have a problem you have found an issue in Joomla's extensions updater I have never run into before (very likely a caching-related issue, since caching seems to be causing it to misbehave). This is very likely to be the case since you are seeing a double update for FOF. In this case you can download Akeeba Backup 8.0.4 and install it on your site from Extensions, Install, Upload & Install.
Finally, please note that Joomla itself is responsible for detecting, downloading and installing updates. Since 2017 we, like all third party Joomla extension developers, are explicitly forbidden from providing our own updater code on pain of being permanently delisted from the Joomla Extensions Directory. If you have a problem with updates you need to report it to the Joomla Project. I have already done that and they seem to ignore me. Maybe if they started receiving a torrent of issue reports they might finally start working on it. Ten years too late, but better late than never...
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!