There's no issus with that.  As long as the source of magnetic attraction is in the north (which on a Flat Map is simply in the center), it works exactly the same way.  Even on a globe, you're curving around the North Pole, so it's really the same difference.

What's more at issue is the Southern Hermisphere, where on a globe compasses shouldn't really work, since they have to be held flat to work correctly even in the North, so how do they work in the South?

So the fact that a compass works in the Southern "Hemisphere" actual militates against a globe when it should not in theory.  What's "distorted" is the fact that compasses work at all in the South.  That's perfectly consistent with a Flat Earth and inconsisten with a globe.

Magnetic fields are 3D and 360 degrees, and follow curves, so the field lines should follow the curve of the earth (north to south). The south pole should behave just like the north except for being the opposite polarity.

Yes, you are right. The compass needle would be parallel and level with the magnetic field lines which curve similarly to the ground as on a globe earth. The needle would be parallel to earth's axis, 45degrees to north at the equator, but pointing in the direction you would walk on a flat earth to reach north. Gravity, which pulls toward the center of a globe earth, keeps you on the ground, and keeps the compass needle centered flat on it's balance needle when held parallel to the ground at whatever location you are measuring from on a globe. It works and makes sense, but is much easier to demonstrate with a whiteboard or animation.

By design, a compass needle is self leveling even if the body of the compass is not perfectly level.

Also, as far as I know, compasses work just fine on the side of hills too, and all sorts of rough terrain.