Just to give an update on this, I've discovered the reason why the banner isn't appearing. I'm now just thinking of the best way to fix the issue.
So, I exported my test project to try and reproduce the problem and here's what I discovered; if you unlock an achievement during a test play (and get the banner), then export the game and try to get that achievement again, you'll get the issue you're experiencing where the banner isn't displayed but the achievement does unlock as normal.
I also noticed that if you unlock an achievement that you didn't unlock during a test play, the banner appears without issue.
I'll keep updating this thread as I go and develop a fix :)