My apologies, I skim-read and thought you were just going to clamp a set of flats on without changing stem or anything else.
Heres a long-winded answer;
Two bikes with identical bar-saddle-bb positioning but with differing toptube lengths WILL handle differently.
On a drop-bar bike:
Riding with your hands on the flats is fine when your sat in the saddle; but try standing up and sprinting with your hands there and it’s much different.
Go out, have a shot and you’ll see. You tend to move forward on the bars the harder you’re riding (your body will drop too). On flat bars you can’t do this so you need to have the bars further out than where the flats-of-a-drop-bar would be, to compensate.
The short Roadrat with a Longer Stem will want to stay travelling in a straight line when your weight is on the bars much more than a long roadrat. The effect happens when you are standing up sprinting, hill-climbing etc. The stem acts as a control-spring wanting to return your bars to the neutral (wheel facing straight forward) position.
Basically, to run flat bars put a longer stem on.