Where do you get ben's badge on gull island on harvest moon tree of tranquility?

first you have to have completed darens rainbow, if you havn't, then do it, after you have done that you need to go over that rainbow to get to gull island where you meet the harvest Sprite named collin.

you then have to complete his rainbow (it is kinda complicated) and cross his rainbow, then you arrive on the other gull island.

from there walk along the beach where a cutscene should appear, you should get the badge in that cutscene...

Hope i helped!!!!