You can add a Category to the custom menu which is great but as soon as you click that link the menu highlights your Blog tab (as opposed to highlighting your category X tab which I argue would be more desirable and logical from a users POV).
Is there any child theme conditional magic that can be done to render a specific category archive page so that it actually appears under the relevant menu tab (and perhaps conversely hide all of those category X items so they don't appear in your Blog either)?