Coming into Season 5, it looks like there is a glitch where players are unable to remove their Back Bling. Epic are aware of the issue and have actually got a fix ready for the next patch but here is a temporary fix for the time-being until V5.1 is released next week.
For PC, removing the Back Bling is actually quite simple. When going into the menu for selecting Back Blings, first click on the ‘X’ selection that is usually used to remove the Back Bling, and then right click twice on any Back Bling in your locker. This should remove the Back Bling on your character.
On console, you simple need to click ‘random’, then quickly press on the ‘X’ selection within the Back Bling inventory. This should remove the Back Bling on your character. You can see this in the video below to help you remove the Back Bling on both PC and console: