Right tool to drill a 3 to 4" diameter hole approximately 3ft deep through remnants of a ground up tree stump
The amount of torque it would take to auger even a 3 inch hole in a tree root would be way beyond what you could safely do with a hand drill. You might be able to rent a gas powered 2-man well drilling auger with as small of a bit as they have for it, then wedge sticks or something in with your post before pouring epoxy resin in it, otherwise you may end up needing a gallon or more of resin.

But...
Anything you do here is only going to last until the tree root fully rots away, then you will have to replace that entire post setup. Once you drill into it, it may accelerate the process and you could be looking at a couple of years max. The "geometry" option seems better to me; just add a post before and after the stump and angle that corner to go around it.

Another option would be to forestall the fence post installation a bit and either use a chemical stump remover (potassium nitrate) now, or drill a hole in the root and burn it out when the weather is dry.