A helicopter should get more stable carrying a load. Not less stable. It is why you don't see big multi-rotor drones or Osprey's on the Bakken oil field carrying 20,000lb bundles of pipe to the drill site. They use S-67's because it's the only thing that flies in a VTOL platform that can do it.
My personal opinion is that there is a bit much FBL/3D mentality here in this tuning procedure going on? The wrong platform has been selected in the first place. A lightly loaded big rotor is "floaty" and inherently unstable. The stock 700 size blades would be much better and more stable, and turn 'em faster. And it is not a matter of having to have a 800 to lift the load. I hold the current record in the Unlimited Class in the RC club in the Beer Lift with a 700 helicopter. Seven 12 packs (64.7 lbs) air lifted and delivered to the "bar" within the time limit to still be judged "ice cold". I beat out a 12 motor multi flown by a DJI NAZA that crashed attempting five 12 packs. The multi was a total loss. We recovered the beer from the crash site, except for two cans that broke open.
You have to have some fun too. It's not all work.
The helicopters with FBL units are TOTALLY unstable trying to do a lift. They are barely controllable at even half their takeoff weight. I don't know why. But Pixhawk ArduPilot handles it with extreme stability and it somehow compensates for it in a way the FBL units don't using basic attitude control (Stabilize). With Pixhawk I have found you can place the load off center and it does not make one bit of difference. That will flip a FBL unit-flown heli right over. I think, but am not sure, that is one of the advantages of using EKF State Estimator for attitude control vs just using gyros, and why Pixhawk handles it so well. Pixhawk seems to automatically "trim" the helicopter out for an unbalanced load, and it does it VERY well. It will fly an auto flight with that 2 x 4 strapped in there just like I showed, and it will have no problems. Just like an S-67 carrying a big bundle of pipe though, you keep the speed down when carrying what is a big load for the helicopter. It doesn't handle as quick, and you don't want it to handle as quick because you're moving more mass around.
No, my P gains aren't even close to .15. I think they're around .06. I am using VFF about .15 though. I will get the current params out of the little 500 and post 'em so you can take a look. I'm flying custom code in that helicopter to get the DDFP tail to work though. The current release code is broken.