Celcuis

Yes, you do need some sort of filter/s in there. I'd leave in the internal filter you have already, and as you already have an air pump going, add one or two of the air-driven sponge filters ( this sort of thing http://www.hollywoodfishfarm.co.nz/product/aqua-one-breeder-sponge-filter-60-19885/) up to it, and then add some sort of filter bacteria (this kind of stuff  http://www.hollywoodfishfarm.co.nz/product/api-stresszyme-1-89l-3/
That should help your water quality no end.

Only a tiny amount that they can eat in a few minutes.

What are your tank parameters ie ammonia, nitrite, nitrate?  There may also have been a small ammonia spike created and adding a form of oxygen displacement will have also helped.
FYI - a fish's stomach is the size of its eye so that is how little they need.  
Water changes will become your friend - if in doubt what is happening and you don't know what to do - do a water change while help arrives

Your fish look "pinch gutted" (which means they have not been feeding) and your water looks cloudy . The latter means your overfeeding has caused a bacterial bloom and the bacteria are using all the oxygen up(that is why the fish are at the surface. You need to remove all the excess food and do a massive water change. Your fish may not have been well when you bought them (they are not showing any colour) or they are now stressed out by the lack of oxygen. Your stocking is low so there should have been plenty of oxygen. You only feed what the fish can eat completely in a few minutes.