Why wouldn't the trailer brake lights come on, when the car lights come on? 1PD makes the brake lights come on just fine.
The issue isn't power usage, it's exceeding the weight limit on the hitch. The Torklift has a max trailer weight of 3,500 pounds (loaded), and 525 pounds of tongue weight. The Draw-Tite has the same 3,500 pound trailer limit, but only 350 pounds of tongue weight. The Hidden Hitch has the same 3,500 pound trailer limit, but only 200 pounds of tongue weight.
Assuming 13% tongue weight, that limits the trailer weights as follows:
Torklift - 3,500 pounds (their max)
Draw-Tite - 2,692 pounds
Hidden Hitch - 1,538 pounds
What I mean by limits is that if you put a heavier trailer than listed above, your tongue weight will likely exceed the tongue weight limits of the hitches. You'll want to weigh the tongue of the trailer (bathroom scale, or bathroom scale with pipes see Google) to ensure you don't exceed the limits of each hitch. Note that if you go to 15% tongue weight, that limits the trailers even further.
If you said "oh, I'll just load the trailer so that more weight is in the back, lightening the tongue weight", that's a bad idea because if you get under 10% you'll find that it lifts the back of the car, reducing traction.
Remember that unlike say an F-150 or F-150L, the car has no anti-sway capability.