Tekonsha P3 will be great.
TM trailers do tow a lot better more with-than-advertised tongue weight. But 700 lbs seems a bit high to me - you could somewhat "adjust" for this by packing heavy things further from the tongue , closer to the middle (or rear) of the TM while getting ready for a trip. I have a super-heavyweight 2619, and I tow with only 550-600 lbs on the hitch.
A slightly more complicated issue is involved in the question of the Lexus "self-leveling" feature versus a weight distribution hitch. Because the Lexus has a big V8 up front, and weighs a lot, you could get by without adding a weight distribution hitch. BUT- and especially with such heavy tongue weight on the hitch, you will be more safe by using a WDH. Without a WDH, too much of the "weight" from the hitch rides on the rear axle of the Lexus, and too little is present on the front axle.
The Lexus feature levels the Tow Vehicle by adjusting the hydraulic pressure within the struts. It does not actually MOVE that weight from the rear axle to the front axle - the rear axle remains with substantial extra weight. In steering and braking, nearly all of the work is being done by the tires of the front axle. In order to do a really good job, the extra hitch weight should be distributed among all 4 Lexus tires - moving weight from the rear axle to the front axle. A WDH creates torque around the hitch, actually moving the weight into better balance.
My own "upper-trim" 4Runner includes a lesser version of your dynamic balancing feature, and my 4Runner is a much smaller vehicle. But, if you did a careful emergency stop (in a straight line) with and without a WDH attachment, you would have much better stopping distance with WDH. The best models of WDH hitches also include sway control, an additional valuable feature for emergency maneuvers.
To summarize: You can get by without one in normal driving, but you will be towing with much higher safety by using a WDH with the Lexus and TM. Congratulations on your new Trailer, and welcome to TrailManor Owner's Forum!