I would contend that "paper over a magnet" model does not adequately explain the problem. We know that the strength of the Earth's magnetic field does not vary at the same altitude at any given point on the Earth's surface. This model fails to explain that.
This claim is testable all on your own, by the way. Take a compass and any magnet for which you know the magnetic dipole moment for said magnet (probably not something you have lying around the kitchen but you get the idea).
You can easily perform this experiment, in reverse, solving for the Earth's magnetic field strength.
http://www.frontiernet.net/~jlkeefer/magstren.htm(EDIT: You can also do this by simply saying your kitchen magnet has magnetic dipole moment of strength 1. Your answer will now be the Earth's magnetic field relative to the strength of your magnet, but the following argument still holds true.)
Take it a step further and use significantly more accurate measurements (rather than looking at a compass use a known current flowing through a wire, for example) and you will find that, upon accounting for differences in elevation, the Earth's magnetic field appears to be a constant.
Why is this inconsistant with Flat Earth Theory? Lets look at two possible models:
1) The Earth is a flat disk floating through space all on its lonesome, with the north pole at the center of the disk and the south pole at any arbitrary distance directly beneath it.
This gives a varying field strength for the Earth because as you move the compass along the surface of the Earth you are moving a different radius from the center of the bar magnet (a somewhat crude approximation, but still accurate enough for this exercise).
2) The Earth is a flat surface that is some part of any other arbitrary shaped surface that has a magnetic field.
This will still give a varying magnetic field strength along the Earth's surface because (for example) if the Earth was the flat surface of a cube (imagine the wall of ice being a circle inscribed in the square surface) the field strength would still change to account for a change in distance from the center of the magnetic dipole moment along the surface of the cube.
Note that this measurement also precludes the possibility of the entire disk being the face of a magnet because this would always yield a field direction of upward whereas here it is measured to be pointing towards the center of the "disk" (according to a FE'er) or towards the north pole of the spherical Earth (according to a RE'er). If you wish to say that perhaps the north and south poles are not on the flat edges of the disk but on the circular edges, 1) You're contradicting your own theory and failing to explain alot of other phenomena 2) You would still measure a changing field strength.