First of all WELCOME to JOF!!!
You hit the number one killer of the standard TT battery. The heating system fan. Not sure as to how many times the heat kicked on over night, but the most, if the heat is not on a lot, would be possibly 2 nights max, with no other loads.
I take it you killed the battery at once already by saying you lost power in 4+ hours. The lowest voltage you should allow your battery to drop to is 12.2 volts (battery 50% rule). A "DEAD" battery is 10.5 volts, not "0" volts.. Some times it only takes one time to REALLY KILL a battery, more than once it is just a matter of a little time. The battery will show a full charge but the rated Ah's (85Ah's on a standard battery) will be reduced considerably, which is what gets you to 4+ hours.
Any type of dry camping needs additional power, especially in the fall, winter and spring when there is a need for heat. You need to monitor your batteries voltage if you intend on getting any life out of them. Like I said 12.2VDC is when you shut down all your 12VDC electronics, and that includes your HEAT. So, based on what you mentioned, you will need another battery... wait now that you have run your existing battery down as far as you did, you will need 2 new batteries. For ANY dry camping, I would recommend (2) 6 volt batteries in series, for 12Volts. A lot of the members use the Trojan T105 batteries, some use COSTCO 6 volt batteries... what ever fits your budget.
Number one get a Digital Voltage display (Amazon) and plug it into your 12Volt accessory receptacle in the TT, easy to watch it that way. Below are a few samples, the prices may not be exactly the same.
Oh, do not use the USB chargers on the Voltage display unless you add more batteries. There is a lot of good battery (and SOLAR) information on our RVing with SOLAR
Before you go out and drop a lot of money for batteries, how do you plan to charge them each day? Generator? SOLAR?
RVing with SOLAR
2013 Jayco Eagle 284BHS
250Watt Grape Solar Panel, MorningStar MPPT 60 Charge Controller
1500 Watt Ramsond PSI, 2 Trojan T145 Batteries (260Ah)
2 - AirSight Wireless IP Cameras (used as rear view cameras)
EnGenius WI-FI extender, D-Link wireless (n) modem
MagicJack Internet Phone
2012 Ford F150XLT, EcoBoost w/3.73
157" Wheel base, HD Towing Package
Our Solar Album https://www.jaycoowners.com/album.php?albumid=329