It will be interesting to see your current tune. We have the same wideband, I leave mine in lambda setting. 0.8 is running rich, 1.1 is lean.
Have you ordered or do you have MPVI 2? HP Tuners is still claiming they will enable Pro features, but it has been years, so I don't hold your breath, and I don't recommend you buy it. If it worked, then it might be worth the extra $$$.
I probably should pay attention to this thread though I'm still very early in learning this stuff, eventually I might have info to contribute. I also have an 06 LSJ, just installed the ZZP stage 2 kit with a 2.9" pulley, adjustable tensioner, 60lb injectors, and have what seems to me to be a decent tuning experience so far. I also have a AEM 30-0300 to install soon but currently only have a recently replaced upstream O2 sensor installed. I have the MPVI2 and ordered the Pro features, connector arrived the other week and my account shows that 'pro' is activated. I haven't had the chance to confirm it'll log but I normally use the bluetooth connection to my laptop while logging data and the pro feature will keep this true whereas a serial connection direct to laptop wouldn't. It's a more expensive route than the parts you linked but it does seem to be a clean install.
I purchased the pro feature set and my MPVI2 through ZZP, here's what I see on my HPtuners account page:
I'm going to add an inline disconnect for the pro connector from the AEM AFR gauge so when I have the MPVI2 plugged in to OBD port I can also connect the AEM gauge to the MPVI2 but when not using MPVI2 I have less wiring hanging around. Hopefully that makes sense, it's a bunch of letters that makes sense in my head but might just be alphabet soup on a screen.