First voyage with 178

Congratulations on a successful 'shakedown' trip in your 178!  We normally dry camp in our 177 and carry a pair of 12v. group size 24, deep cycle, Interstate batteries with us - they are now 9.5 years old and still holding up well.  First thing - purchase a volt meter or similar, I purchased mine from Harbor Freight years ago, cost was $ 4.  As Vann stated above don't run it below 50% S.O.C. (12.2 volts) before charging or swapping out to another battery.  Last year this time we were in Colorado and Utah for a month.  Even when the night temps in the Rockies were 50 or below-which they were every night, sometimes hitting 39, we were able to get 3 to 3.5 days out of one battery before switching to the other.  The furnace was the big culprit in battery usage.  In summer time we can go a week on one battery and still be sitting at 12.3 - 12.4 volts.  That is running L.E.D. lights, fridge, stove, HW on propane, and water pump on 12 volts and minimal use of the FF fan in the bathroom. 

Battery maintenance is crucial for long life.  As stated above don't run below 12.2v, keep battery and posts clean.  Once or twice a year I make a solution of baking soda and water and brush all over the battery and flush off with a water hose.  Keep an eye on electrolyte levels, keep full using only distilled water, and when not camping unhook the ground of your battery so the parasitic drains don't drain it down to zero or install a battery dis-connect. 

Your Lexus is the hybrid, right? Pretty much the same vehicle as the Highlander Hybrid.  They only have a 3500 lb tow rating with a 350 lb tongue weight rating. I would not recommend towing a 178 with that, especially not one loaded for dry camping with water and batteries. That is very marginal for a 178, especially if you dry camp and carry water and an extra battery. I own a Prius so love the hybrid drive but when I bought my tow vehicle it was a 2012 V6 Highlander with tow package, not a hybrid. 

Please do yourself a favor and weigh your fully loaded vehicle, trailer, and trailer tongue to be sure you are in spec for your max combined vehicle weight, tongue weight, and trailer weight, and axle weights (which are all either on your driver's door sticker, your vehicle manual, or on the trailer sticker). If you want I have posted a procedure for doing this at a public scale which you can find using the advanced search function. If you can't find it let me know. 

My published 179 empty weight is 2630, but that doesn't include any "optional" stuff, or even a single battery. Loaded for boon docking (dual golf cart batteries, full fresh water tank, food, etc) it comes in at 3700. My tongue weight is maxed out right at 500 lbs. And we're not packing heavy at all. Weight adds up very quickly, and the listed trailer empty weights are optimistic. On the tow vehicle side tow ratings are listed assuming nothing is in the TV except a driver. Also highly optimistic.

Bottom line, don't guess. Weigh your fully loaded trailer and TV with gear, full fuel, and passengers, I think you'll be surprised. Check all your axle weights, trailer weight, tongue weight, TV weight, and combined vehicle weight. My 5000 lb rated Highlander is OK, but I wouldn't add any more weight to my rig.  

Here is a nice calculator to load all your numbers into, it also calculates the effect of adding a weight distribution hitch. The wdh does not give you more towing capacity or axle capacity, just rebalances the load distribution. 

You can weigh at a public scale or sometimes for free at your local waste transfer station. You have to go through the scales twice, once  with and once without the trailer. Do not have your weight distribution hitch tensioned. Get the weight of each axle by adding one axle at a time on the scale and then subtracting. When you go through without the trailer leave the wdh in the receiver. Subtract TV weight from total rig weight to get trailer weight. Subtract TV weight without the trailer from the weight you got for the two TV axles with the trailer attached to get tongue weight.