I have not verified how old the saying "den Bogen raushaben" is. Maybe the origin is connected with weaving. My association was always skiing. A beginner has to learn how to manage to go into a curve. When he knows how to do it then he might say: Jetzt hab ich den Bogen raus meaning Ich habe herausgekriegt, wie man einen Bogen fährt. But I would consider this explanation not as a historical explanation, but only as a plausibility explanation that is finding a situation where the saying might make sense.
Often there are historical explanations but they are difficult to verify or they do not convince. Then a plausibility explanation at least helps to understand the saying.
At any case I would not think of a weapon as a bow in the pocket. You wear a bow on your back and if need be you take it off your back, but not out of your pocket. That is an association that may be helpful for someone but the image in not correct in details.