Akeeba Backup 4.7 is compatible with Joomla! 1.6 and all the way to 3.4. I tested that personally. The only solution we have for you is upgrade to 4.7.
Why you don't see the settings? I can think of two issues. Either some old files are not being removed or you have a Javascript issue. I am pretty sure about this since the jQuery-based code that renders the Configuration page was unchanged between December 2009 when it was first written and early 2017 when Akeeba Backup 4.7 was released. We only changed that code in Akeeba Backup 5.3 and later versions, removing the dependency to jQuery. So, what you have (4.1) and what we tell you to install (4.7) use the same code. The only difference would be what I mentioned.
I'd first start by exporting the backup profiles and keeping a copy of all the backups you have stored on your site just in case.
Then I'd recommend deleting the administrator/components/com_akeeba folder (except for the /backup subdirectory). Then install Akeeba Backup 4.7.
If that didn't help, check the browser's Javascript console. What is the Javascript error? Do you get any 403 errors trying to load a Javascript file? Do you have any plugins or a CDN, like CloudFlare, which tries to rearrange the script loading on the page (our code will NOT work if a third party tries to make our dependencies load after our page script initialization which uses those dependencies, of course).
If all else fails ping me so I can make this ticket private and ask you for connection details.
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!