After learning so much from this forum, I wanted to contribute back. So this exact discussion is perfectly timed.

Just back from Prague on an avios ticket. When I did online check-in via the app, the boarding pass on the app does say 'Saver' fare but also has 'Carry-on bag' which is all we needed.

So no need to stress, you're good to go straight through security with the carry-on bags.