Welcome to the forum Ken. You've gotten some good advice above and it's all spot on but I'll give you my experiences and my equipment and my thoughts.
2005 Jayco 27BH trailer, two 6 volt batteries from Costco and a Honda eu2000i inverter generator. 40 gallon fresh water tank, 36 gallon gray and black tank, 4 kids and my wife. I boondock in the desert almost 90 percent of the time I'm camping.
My family will go through the 40 gallon tank of water in three to four days, and that's using the water sparingly. I have all boys and am in the desert so they have to pee in the bushes during the day. But they still need to wash their hands and brush their teeth from time to time. And I need water to cook and put out the fire at night.
My trailer is a hardsided trailer, not a popup and I run the furnace at about 65f at night (mid to high 30's outside) and that hits the batteries hard. I can make it two nights easy before recharging if I want. But I have a generator that I paid good money for so I run it.
My brothers popup though runs the furnace almost constantly all night and he needs the recharge (off my generator no less)in the morning. More blankets or better sleeping bags would probably help there.
Enough about me, now about your question.
#1 forget anything about using an inverter for comfort in your trailer, you and I just can't carry enough battery for this.
#2 Plan on 1 night of heat per battery installed. Yes if you keep the thermostat low you can usually scrape out a second night, but it's the fan on the furnace that's chewing through your battery storage.
#3 you need a way to put electricity back in your battery. I use a generator but I don't like the noise. The Honda is one of the quietest generators on the market. But to not hear it I need to walk about 75 - 100 yards away in the desert, so you will hear the noise in camp. But it is relatively quiet, almost like a low hum sitting on the other side of the trailer from the chairs. I charge the phones and things like that during the day when the generator s running.
#4 Solar is silent, but only works when the sun is shining and you need more panels than you have roof space to charge your battery up after a night of furnace.
#5 In a pinch you can turn your vehicle around and use some jumper cables to charge the trailer batteries. But that's a really expensive battery charger. :)
#6 Don't count on your tow vehicle charging the batteries while pulling the trailer. It takes many hours towing to put a decent charge in the battery.
If you are already a tent camper, just take your gear minus the tent and go camping. Use the trailer like you want to and see how it works. If you burn through the battery charge in one night you still have your trusty camping gear and a big comfortable trailer to sleep in.