Spent time between two campuses, one isn't degree granting. How do I list this on my CV/Application?
As your professors suggested you can just list X University. This is the university that grants you your degree, it does not matter at which campus you studied. Since X University is willing to associate with the branch campus so strongly I would assume the campus has a similar reputation and as such I don't see how anyone would be deceived there.

At the same time I don't think it would hurt your application to mention that you studied in two countries and experienced two cultures. Having experience dealing with different customs may smooth your interaction with researchers from other cultures. You can for example make a brief mention (one or two sentences) in a motivation letter. Of course you should consider carefully whether there are more important things you could mention in those sentences.

If you insist on including this in your CV make it clear that it is a single degree. My personal preference would be to have this in a single line, for example:

X University .... City A, State and Campus B .... City, Country B

Regarding your transcript: Don't submit the same transcript twice. It will only lead to confusion and likely the assumption that you submitted the wrong transcript for one of them.

Regarding your publications: Don't worry about what affiliations are on there. It is not unusual to have affiliations to different departments of the same university on research work. Affiliations on your publications carry no weight in who granted you your degree.