Laminate floor against fireplace brick

We are installing a laminate floor in our family room and have precisely the same problem as described in this question. The brick is not even close to flat where it meets the floor, leading to ugly gaps.

The solutions given in the linked question are not what we’d like to have. No trim or border. We’d really much rather have the "floor underneath" look, where the floor appears to simply disappear underneath everything in the room, including the fireplace brick.

This implies that we have to cut the brick somehow. I’ve seen a few online videos showing how to do this, but they were lucky enough to have a grout line at floor level. There is no grout line at the correct level – the floor goes right into the physical brick.

I am looking for methods and suggestions regarding how to best cut the brick. Tools, gotchas, ways to make sure it looks neat, etc. I only get one shot at getting this right. 🙂

As an alternative, I’d consider using some kind of a "gap filler" to hide the gap, as long as it is level with the floor and can be colored to match either the grout or the floor.