Is there some trick to defeating the dog in the estate or is it simply a matter of grinding your doggo?
It's grinding, but also equipment.
You won't beat it without Recoup, your dog's self healing skill, there's simply too much damage going out and he has too much HP.
But with the right loadout, you can help mitigate a lot of the pain. Stuff like the 5% HP regen, SP regen, health and TP augment, as well as the Follow-up attack item will make it much easier to deal with.
You can also make the earlier turns easier by stacking all your available buffs, so the dog's three skills, Priest's three boosts and passive regen, and Evil's Wide Barrier and Magnify.
If/When you do win the fight, your screen is gonna fade to black. It's a bug that was supposed to be fixed, but as of 2.05 it's still here. You can still move around though, it's only the screen that's blacked out, and it'll come back if you transition to a new area.
So, admitting you were facing the dog from the left : walk a few seconds to the left, then take two steps down, nine steps left, two steps up, and run to the left, you'll reach the area transition.