What you are suggesting is that there is a quantization law for magnetic fields, so that if the field is too small, it is actually exactly zero. This is not true in quantum electrodynamics, the field shrinks to zero as a power law, and the influence of the magnet is felt arbitrarily far out.
But the experiment which you need to measure the field becomes larger and larger. Given a particle of charge e, you can take it in a loop around the region with a magnetic field, to see interference fringes which depend on the area of the loop according to the phase law
$$\delta \phi = \oint A dl = \int B dA $$
To get an equal phase change, you need to make a bigger loop when you are far enough out.
The question of the measurability of tiny fields is not purely academic. The phase method of detecting fields is extremely sensitive to tiny fields when you use a superconducting loop and measure magnetic fields by the phase the current gets around the loop, this is a SQUID. The SQUID can measure fields which would be too small to measure other ways, and it can be adjusted to measure fields that are infinitesimal with regards to less sensitive detectors.
The philosophical position that all real quantities must be eventually discrete is not particularly useful, because the real quantities can arise from large-system-limits, and so be arbitrarily fine-grained as you get to larger and larger system sizes. For example, if the temperature is arbitrarily small, is it indistinguishable from exactly zero temperature? This depends on the size of the system. If you make the system bigger, you can see the difference from zero temperature even finer.
The limit of large number of photon exchanges, in a Feynman particle view of the field, is the large number limit that makes a tiny field make sense. If the field is too tiny, you will only get a finite number of photons affecting your device, and the effects vanish. But if you make the device bigger, you become sensitive to more photons. These type of large N quantities can philosophically be real valued without any contradiction, because the grain-size for the real-number quantity is physical, and determined by the parameters of the system and measuring device.