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Call your host and ask them to check the maximum number of POST variables. If it's 100, ask them to increase it to 150. Right now, the Configuration page has 107 variables across all features you can configure (less than half are shown on the screen at a time, but all of them are in the form). That's why you get the HTTP 406 error on servers with a very conservative 100 POST variables limit.

In the meantime, I created a workaround. Please download and install this dev release: https://www.akeeba.com/download/akeebabackuppro-dev/9-9-3-dev202405010941-revcdc2ccc.html I believe you'll find your problem has gone away.

As I mentioned earlier, the problem is that there are just over 100 individual parameters in the HTML form which is being sent to the server when you save the configuration of a backup profile. There a few servers which have a limit on the maximum number of POST parameters set to something equal to or lower than 100. This is why saving the configuration fails, and why sometimes saving an article in Joomla! fails if you have dozens of user groups (each user group adds a number of separate parameters in the HTML form).

Modern browsers -- released in the last 5 to 6 years -- have a new feature which allows us to serialise a form as a JSON document. In other words, instead of having to submit 107 individual parameters we only need to submit three: the task (save & close, apply, save & new) you asked Joomla! to perform, an anti-CSRF security token, and a big JSON document with the other 104 parameters of the form. Since the big JSON document counts as one parameter and is small enough (just a couple of kilobytes) so as not to trigger any maximum POST size limits the save goes through without getting an HTTP 406 error.

Please do let me know if that works for you.