How can I make Bluetooth (2.4 GHz) reception area narrow and controllable?
Neither will work.

If you build a metal case with a narrow opening, you'll have an even wider beam; the smaller the aperture, the larger the beam.

It's practically impossible in this form factor to controlledly cut off the existing antenna, and still have something to attach an external antenna to.

But even so: at 2.4 GHz, the ground you mention will work as an OK reflector, and hence, things will still work in the area "indirectly" illuminated by ground reflection.

2.4 GHz devices (such as Bluetooth devices) have to be designed to work with multiple reflections, so it being impossible to limit the area if you're illuminating a wall or a piece of ground is a feature your device has to have.

You could try to add so much attenuation to the antenna that the signal is really really weak, but since a reflection might have relatively little loss compared to the first couple of meters of free space loss, this won't work out, either.

Long story short: you can't.