Thanks gents.
I had not thought of a grout bag. Sounds like the best way to fill in the mortar (though I still plan to butter the hole and the brick before I insert the brick).
I have a 4" grinder, had not thought of that. Must check whether I have a diamond blade, but that's easy. In...
My two-flue chimney (3 bricks x 4 bricks) has half a dozen flaking bricks. The cause is either bad bricks (29 years old) or water penetration from the cap (fixed). All the bad bricks are in two rows, about six feet below the top.
I have removed one corner brick. This took about 45 minutes...