You can change costumes on "Combat Equipment" screen by pressing down the R stick on your controller.
About "pressing" the L/R stick (Nintendo Switch):
In addition to tilting like a lever, the stick can also be pressed down like a button.
Pressing the stick straight down until you hear a "click" sound is called "pressing the L (or R) stick."
Note: This operation is different from "pressing the L (or R) button."
For the locations and names of buttons/sticks on the Nintendo Switch controller, please refer to Nintendo's support page: "Joy-Con Controller Diagram."
If DLC costumes do not appear
Please follow the steps in the FAQ: 【Switch】DLC is not applied to the game.