Which bonfire I must use Bonfire Ascetic to make the Doors of Pharros red phantom invader respawn

I tried for a good while to find a way to get back there. No matter what I tried, though, I couldn't find a way to get back over the rubble. So, unless there's a really well-hidden trick, I'm inclined to say that it's not possible.

As consolation, however:

if you keep pressing onward along the outside path, there's another bonfire not very far away at all.

No, it doesn't affect the entire named area. It only affects the area between the bonfire on which the ascetic is used and the next bonfire. All enemies after the next bonfire will not be affected. For example, the first Aldia's keep bonfire will affect the rats outside the keep, the black phantom within the keep, the mirror knight enemies, the ogre and the giant basilisk but will not effect any enemies beyond that point as they all fall within the vicinity of the second bonfire, despite all the enemies beyond that point still belonging to Aldia's Keep. If you used an ascetic on the second bonfire, then all the aforementioned enemies would not be affected, and only the enemies between the second bonfire and the boss (including the boss, as the boss is 'associated' with the second bonfire) will be affected. I know it's a poor explanation, but hopefully it gave you an idea as to how the mechanic works.