With the palm down, the brachioradialis is on the inside of the arm, not the outside. The flexor carpi ulnaris is still a flexor and can be aggravated by curls. Doing hammer curls activates the flexor carpi radialis more than anything. However, bar curls require activation of all the flexors, especially with the unnatural lateral rotation of the forearm. Since the flexor carpi ulnaris is a medial rotator, I'd imagine that the lateral rotation results in great activation of the muscle and therefore extremely high tension on that muscle specifically.
I'm also mentioning this based on personal experience. Doing straight bar curls aggravates the inside of my wrists significantly. This sounds like a similar problem, especially due to his description of pain location. Either way, a PT could ID and prescribe exercises to fix it. Us identifying it doesn't help so it's just futile anyhow