Welcome to the forum!
You may need to use a diode or relay. It's possible your cube lights are too much of a draw for the factory circuit.
For the backup lights on my Sport S without factory LEDs, the white wire with brown stripe is for reverse bulb 12v +.
The lights I added are 2 x 18w and...
The LCAs are 1/4" longer than stock and will add caster that the JLs need to improve the steering tracking and stability.
The Rubicon suspension and mopar lift LCAs are one of my best mods to date. I also added front spacers to get rid of the rake.
Just completed the project and very excited for the results at night!
Here's a daytime picture below. (For reference, the backup bulbs in the taillights are aftermarket superbright 6k LEDs.)
I'll snap some night-time pics also.