If it has been like this since you owned it, and never noticed it, this is a possible scenario:
The headlights could be programmed to have the high beams be the daytime running lights,(DRL), at reduced voltage.
So, when you put it in gear, the high beams should come on.
When you turn on the regular headlights, the high beams go out, and the low beams take over.
Then, when you select high beams, the low beams go out, and the high beams come on.
If there is a bad connection for the passenger side high beam, it will be out when in DRL mode, and when you select high beams.
I would look for voltage at the passenger high beam connector pin with high beams on. If you get voltage, check for continuity from the ground pin in the connector to the body.
If you have an aftermarket program installed, I believe the dealer can flash everything back to stock, based on the VIN. Call and ask so you know in case you need to apply that option.
You said earlier you know it's not the bulb. Is that because the LEDs behave the same way as when you have the OEM bulbs installed?
2008 JKU Rubicon