e.g. having a source of neutrinos 1 km away from the receiver, and see if they still detect it 60 ms "before" (perhaps even before they fired them)

Is the cost/technology for doing this kind of quick check prohibitive?

I was wondering if the neutrino detector does have to be directed precisely towards cern or it's omnidirectional. Does anyone know?

A neutrino detector is omnidirectional. It would be almost impossible to make it directional even if you wanted to.

You can try to measure which direction the scattered particles from a neutrino collision travel. But the actual detector is omnidirectional.

But all that said, a strong neutrino source isn't easy to just do.

assuming the emission is also omnidirectional the required strength decreases by square of distance, so it shouldn't require a strong source for 1km.

I guess it's more difficult to tell exactly when the bursts are being emitted.

No, the emission is directional.