I've done bear and hog both, but the process I've found to be most efficient is this:
Take the clean, white fat and cut it into chunks. I prefer to run it through a grinder. It renders quicker and more thoroughly this way. I put it in a cast iron pot and turn the stove on low heat. You don't want to cook it, just melt it. It may take a few hours, but the fat will slowly turn to liquid. At first, it's very cloudy but as you continue to render, it will clear up. I take a slotted spoon and scoop out any solids that are floating. Other impurities will have settled on the bottom. Everything else is pure bear grease. I've used a large syringe (for injecting turkeys) and drawn out the clear liquid and then deposited it into clean jars. It will set up over night and make a pure white grease like Crisco.
We keep ours in the freezer until we need a new jar, and just keep it in the fridge to use as needed. It makes awesome biscuits. I used bear grease to season my Blackstone. It's light and clean, and if you did it right, it has no off-putting flavor or odor. You can put your nose in the jar and can't smell a thing.
I have no problem with bear dogs being legal, as long as there was a season you could hunt them without dogs being legal. I see bears all the time during bow and ML season (before our bear season opens,) but after the first dog is turned loose opening morning, the bears pretty much go nocturnal and stay that way the rest of the season.
That's my issue. It renders ground hunting for bears a fruitless exercise. In NC, if you don't hunt with dogs, you ain't killing many bears. If the seasons were separate, I might be more inclined to support it.