A few thoughts/questions for the OP -
Does your hitch have an anti-sway function. I usually think of those side sway damping things as a bandaid for poor trailer dynamics, but I did notice a significant benefit when towing a full size box with my 4Runner. Travel trailers are about the worst trailer you can get - high polar moment, high side windage, etc. Towing with a midsize SUV isn't an ideal match. I've never used or felt like I needed that sway control on full size trucks with the same trailer, but it really does help on short wheel base tow vehicles like the LX.
Not much help on transmission tuning. It is unfortunate that it doesn't have a tow setting because the software for that should already be 95% complete on the Sequoia. May even be just a copy and paste of the code over to the LX. It might be worth a look at a Sequoia to see if that's a better vehicle for you if you're towing a lot. I have not towed with a 3rd gen Sequoia - but reviews seem to be generally positive and it has most of the features you're wanting.
Also question about the exhaust trailer plug location - doesn't the hybrid on the LX mean that the engine should be shut off when it's sitting in Park? Is the engine running when you're down trying to plug it in? One reason I don't love the location underneath is that I really like the Curt Echo brake controller - I have one and use it for all of my trailers including use it instead of the OEM controller in my Tundra because it works better. The downward angle of the underneath plug tends to be prone to the Echo device falling out if you don't make sure to strap the plug lid in place to make sure the retainer clip keeps it plugged in. I also found in my 4R with similar location that I occasionally needed a small jumper to extend the trailer plug to reach.
