I've created a replica of the site.
I've uninstalled ALL extensions other than CB and CB Plugins.
I've uninstalled all non-core extensions (plugins, modules, etc.)
I've cleaned the htaccess file (from all the 'security' settings added by Akeeba Admin)
I've even uninstalled CB Paid Subscriptions plugin and re-installed.

Still, having the same problems (changing access to CB Subs redirects to Control Panel and doesn't change the settings).

On a clean install with just CB/CBSubs the problem doesn't happen (spin still happens, but the change IS saved), just on this specific site.

I'm running out of things to test/track down.

Is there any possibility that a configuration within CB (or CBsubs) might be causing this? (Or a CB plugin)?

I think I MIGHT have narrowed down the issue.
The site has a lot of custom admin menus (using the new admin menu features in 3.7). After deleting all those custom menus our dev site seems to be working as expected.
Now to just narrow down and isolate which menu is causing the issue

Thanks
That's what I did on the previous round; however, I did it after uninstalling everything else.
On a new replication of the site, just removing all the custom menus doesn't resolve the issue, so it must be multiple issues causing the problem.
I'm not going in 'reverse' order (having removed the custom menu items) to try to identify the 2nd conflict.

I do agree that this looks like a routing issue.

I would strongly encourage/request/plead that you make CBSubs permission settings work with the new custom admin menu items (as the core CBSubs permission settings already seem to do).

I'll post back here as soon as I track down any other issues and get this identified.

Well, completely deleting all the custom menu items (and menus) didn't resolve the problem.
Disabling all the non-core (other than CB) components/plugins/modules didn't resolve the issue.

The only things left (that I did previously but didn't do yet) are to completely uninstall those disabled exetensions (which I shouldn't think would be necessary) and to clean out the htaccess file again.

I'm guessing that the htaccess is the issue (because it can effect the routing); however, I'm growing concerned that CB's core permissions settings all work but CB Subs don't. Seems to me that there is a problem (or strangely different) way that the 2 different systems are working (or interacting with various routing issues).

If you know of an htaccess setting that is known to cause problems with cbsubs that would be helpful to know. Otherwise I'll try to debug that manually here.