Okay... got a little testing done. So far, here's what I have.
With a 12-volt battery, the brakes actuate fine on three of the four wheels. A compass held near the wheels confirms that the fourth brake doesn't seem to be activating at all. That is new information, as that is not the wheel that I was doing my troubleshooting on previously, so there may be more than one issue here.
With the manual activation lever on the brake controller pressed, three of the four wheels lock up strong. That is not what was happening previously, but one difference is that previously, I was using a clamp to hold the lever, while this time, my girlfriend was pushing the lever, so... I don't know... maybe the clamp wasn't holding it all the way in or something, although I think it was... but who knows?
With the brake pedal pressed, there was barely any activation, but I have recently learned about the existence of inertia-based brake controllers, and if I have one of those, then that would explain why the brakes aren't activating sitting still--in which case, I apologize for taking up y'all's time, and I sincerely appreciate the help with the troubleshoot (it's been educational!), but I may have totally mis-diagnosed the problem. My last truck's controller wasn't intertial, and it didn't matter whether you hit the brake pedal or not. In fact, it didn't even have a manual activation lever. So there you go.
But the silver lining is that I believe I've discovered that one of my four brakes isn't activating, so that's something. It's a good thing I have brakes on all four wheels, because otherwise I'd have zero braking on the left side of the trailer!
I've got a call in to the dealer to confirm that the controller is inertial, but it seems like if the controller grabs hard when you press the lever, but not when you press the brake (standing still) that's the most likely explanation.