Could an asteroid knock the moon out of its orbit?

Basically no and not by a long shot. The asteroid's measurements are pretty vague, but lets give it a nice clean 600 meters in diameter and the same density of the moon. The moon is 3,474 km in diameter or 3,474,000 meters. So the moon is nearly 5,790 times the diameter and that gives is 194 billion times the mass as that asteroid. Conservation of momentum, if the asteroid transfers all of it's 78,000 mph into the moon, with no energy going into angular momentum, 78,000 / 194,000,000,000 = .0000004 mph. That's roughly the speed your fingernails grow.

To knock the moon out of orbit, it's velocity would need to be increased by oh, very rough ballpark, maybe 400-500 mph. The additional velocity needed for the moon to reach escape velocity from the Earth is .414 x 2288 or about 947 mph, but given that it has an elongated orbit and that it would escape the Earth's sphere of influence well below escape velocity, maybe as little 400 extra MPH could knock the moon into a far enough orbit that it could escape the Earth.

Figuring 78,000 MPH, and you want to move the moon some 400 or more MPH with ah it just right, you'd need an object about a 5th or 6th the diameter of the moon or 600 KM in diameter give or take, assuming it was straight on impact as a glancing blow would lose a share of energy into angular momentum and assuming it hits the moon away from the Earth, not towards as towards would require even more energy.

This space pebble doesn't even come close. You'd need something the size of a small dwarf planet to have a chance at knocking the moon out of the Earth's orbit and that speed.

300-600 meters is huge, if it hits the Earth. That's probably a few square miles of earth practically leveled to the ground and broken windows and knocked down trees a good deal further than that. If it hits the moon it would leave a nice crater, maybe a miles or two across but that's about it.

It seems like you're only talking about the moon being kicked out into
space. Would the same amount of energy be needed to knock the moon out
of its orbit in towards Earth?

The mathematics of crashing is a bit more complicated. Escape Velocity vs Orbital Velocity is easy, just straight multiplication. More details here, and because the Moon wouldn't need to reach full escaple velocity, I cut the number in half - and it might be a bit less than that.

Moon's orbital Speed (2,288 MPH) x (Square root of 2 minus 1, or 0.414).

Crashing the Moon into the Earth - a bit harder. There's probably a simple enough formula, but I'm not sure what it is, but I believe that would take even more energy.

It's discussed here without math and here, with math. But it would take a huge amount of energy, probably 3 or 4 times more energy to push the Moon into the Earth than to push it out of orbit if I was to guess.