I have done a lot of winter camping without hookups. I use an unventer propane  radiant heater like an Empire or Comfort Glow. They are not approved for RV's but many people use them. The manufacturer's web sites explain how they work. They are low presure LP and need a regulated supply.

Also, Portable Buddy  Heater is a new low cost heater said to be safe to use and can be hooked to an unregulated LP bulk bottle. Camping World and many others sell them for about $100. Be sure to read all about them. There are catalytic heaters approved for RV's made by Olympian. These are probably the best but are more expensive.

If you run the fan on the regular heater your battery will be dead before very long so the other heaters keep you warm without killing the battery.

As far as the waste water goes, I believe the black water will be ok as long as the interior of the trailer is warm. The book recommends adding rv antifreeze when you charge the toilet and I certainly would do that. The grey water can be drained to a sump if allowed in your area or captured in buckets outside of the trailer to keep the exposed grey water tank from freezing. Or you could add some rv antifreeze to the grey water to keep it from freezing.

Enjoy your winter camping it is great fun.