Well, an occasional shooter can probably do better with a gun he is familiar with than a dedicated trap gun that he is not.
Having said that, I can't answer whether it is "worth it" to you to buy another barrel....your barrel will certainly hit targets if you point it correctly. I know a fair number of good shooters who shot a long time with their field guns.
The main difference in a field grade stock and a trap stock is usually the drop at the comb (where your face goes). A field gun will generally have a lower stock (more drop) than a trap stock (less drop).
Field guns and stocks are generally lighter and made to be carried more and quickly mounted and shot, hence the more drop of the stock for an easy quick shot.
A trap stock is usually nearly parallel to the bore of the gun and when mounted you would place your head down a little more firmly for a consistent sight picture. Not always appropriate in a field setting when the game is flying away from you. Hunters usually find that a trap gun feels "uncomfortable" to them at first.
Trap stocks, when fitted properly, help to lessen the felt recoil to your face.
You mention that this is for "occasional" trap use and I would offer that neither a different stock or barrel will contribute much to your initial skills, but practice will. A vent rib barrel is pleasant to look down, but not at all neccessary to shoot decent scores.
If you get the "bug" to continue, try a few other guns first before spending any un-neccessary money on the 1300. Save your $$ budget for practice.