This will have to be done on your second (or third, etc.) play through the game. After you finish the game, start an EX New Game and buy the perk "1/2 experience" from the Grade shop, along with anything else that will help such as the HP/TP increases, items, artes and skills. This way you can be sure that you do not exceed level 15 before getting to Barbaros but will still be well prepared. You also can not level past 15 with the experience earned from beating Barbaros, so make sure you're level 14 when you get to him to be safe. Switch the battle difficulty to Easy from the data tab in the menu to help since you will be a lower level than the bosses. Try to avoid most normal battles as well. If you are toward the top of Ghasfarost and are not level 14 yet, go ahead and level for a bit to get there so you can have all the health and attack power possible.
Note: There may be a miswording of the achievement and you may have to actually be below 15 to get it, not at 15. Aim to be level 14 after the experience earned from the boss just to be safe.