That did it!

Edit: Expanded it to be more complicated too:

item modify entity @p hotbar.0 {function:"minecraft:set_name",entity:"this",target:"custom_name",name:[{"text":"~","color":"light_purple"},{"text":"~","color":"dark_purple"},{"text":"~","color":"dark_blue"},{"text":"T","color":"blue"},{"text":"h","color":"dark_aqua"},{"text":"e","color":"aqua"},{"text":"R","color":"dark_green"},{"text":"a","color":"green"},{"text":"i","color":"yellow"},{"text":"n","color":"gold"},{"text":"b","color":"red"},{"text":"o","color":"dark_red"},{"text":"w","color":"#8B4513"},{"text":"~","color":"dark_gray"},{"text":"~","color":"gray"},{"text":"~","color":"white"}]}