You'll be ok with the P2, but the P3 is far superior. There will be a difference in stoping control.
You'll like the TRD sway. Noticeable cornering and sway control.
I put Bilstein rear shocks on mine after I blew a stock shock. That happened while towing over a "bouncy" highway. I always take it easy on that particular stretch as a weird "resonance frequency" bounce starts if over 45 mph. Only happens when pulling the trailer. Funny thing is that it won't stop until my speed drops below 35 mph but won't start unless over 45. Anyway, I forgot about it and hit it about 65 mph and couldn't get slowed down fast enough. Thought I was gonna break the frame or hitch. Saw the leaky shock a little later when we stopped for fuel. The Bilstein shocks help a little with that bounce. They really make the truck ride stiff.
One other thing that you might consider is getting a LT type tires for your Tundra. I used the stock P tires until they were shot then upgraded. That made a huge stability/control improvement.
2000 Cherokee 31BH, "flipped axles and raised",LEDs, solar, inverter, etc..aka "boondock ready". '07 Tundra 5.7L DC-LB, modded as well.