Properly adjusted brakes should be able to skid all four tires on dry pavement when the voltage/ gain/ power is cranked all the way up. Something is wrong and you need to fix whatever is amiss.
If one brake is faulty, say due to contamination from a leaking grease seal, then the other three should still lock up. You may have a wiring issue like a poor ground. Adding an additional ground wire between the trailer frame and TV can dramatically reduce or prevent brake and lighting problems. The vast majority of trailer brake and light problems are due to poor grounding, and this is the first thing to look for when things are acting strange....
I have not read anything here about the quality of the Oliver chassis wiring, but I expect that it is FAR better than that of the typical Harbor Freight utility trailer or most travel trailers. But that doesn't mean that there isn't a loose or corroded connection somewhere.
eTrailer.com has lots of great troubleshooting articles like this one .... https://www.etrailer.com/faq-testing-trailer-brake-magnets-for-proper-function.aspx
Unless you have serviced them very recently, it would probably be a good idea to physically inspect and adjust all four brakes, just to establish a baseline for further troubleshooting. A decent digital multimeter with ammeter function and some advice from a knowledgeable neighbor can really help if you have an electrical problem, or just ask here.
BTW, the P3 is a great controller. I have one in my Ram and it works very well. It is very popular for many reasons. If you do not already have one, I _strongly_ suggest you buy one of these special Tekonsha testers, ASAP, to use for testing both the tow vehicle lighting circuits and the function of the controller.
It serves as a dummy load so you can activate the features of your P3, including the manual over-ride lever. (It eliminates the No Trailer warning when the trailer is disconnected.) It may save you much time and confusion, and your wife will not yell at you for saying bad words.
Good luck.
John Davies
Spokane WA