Ok, after about 4 hours of testing, I've finally worked out what the issue is. Whatever colour I make the AE bg, it doesn't transfer to Premiere.
So what I was doing was using a white bg with black text. In Prem, it was not showing a bg and just making it transparent (showing the default black), so that's why my text wasn't showing up - it was black on a black bg.
So my question now has changed: How can I get the background to not be transparent when I import it to Premiere?