Maxwell's equations are not paradoxical at all and they are perfectly in accordance with relativity. That's how relativity was discovered: The Maxwell equations, although known to be very successful in discribing electromagnetic phenomena, where not Galilei invariant. Many physicists (including Voigt, FitzGerald, Lorentz, and Poincare) thought one had to introduce a preferred frame of reference (called the "ether rest frame") to accommodate this. The more they thought about it, the more complicated the material named ether became, and finally it was Einstein's fresh point of view that one has to adapt the space-time description to the invariance group of the Maxwell Equations, now called Poincare symmetry (invariance of the natural laws under proper orthochronous Lorentz transformations and space-time translations).
The usual treatment of macroscopic electrodynamics is, however, mostly plagued by tacitly making non-relativistic approximations to the treatment of matter and thus the consitutive equations. This is a pity, because it makes a lot of unnecessary trouble with interesting phenomena like the homopolar generator and energy-momentum bilance, leading to complicated explanations with "hidden momenta" and all that. If one treats everything relativistically, no such oddities are necessary.
Another source of confusion is that many textbooks state the Maxwell equations in integral form making (again often tacitly) special assumptions. Particularly Faraday's Law is plagued from these sins. When dealing with time dependent surfaces and its boudaries, one has to include the magnetic force in the electromotive force, and everything is fine. It's very nicely explained in the Wikipedia:
http://en.wikipedia.org/wiki/Faraday's_law_of_induction#Proof_of_Faraday.27s_law
The most simple form, is naturally the differential (local) form of the equations, because classical Maxwell theory is the paradigmatic example of a relativistical (classical) field theory.
Now to your problem: When dealing with this problem, you have to solve first the problem of the electromagnetic fields produced by the time-dependent current in the larger loop. Here you have to use the full time-dependent Maxwell equations since you are dealing with a situation where the relevant parts of the setup are much larger than the typical wavelengths of the produced em. waves. The electromagnetic fields cannot propagator faster than the speed of light in vacuo, and thus the front of the signal reaches the smaller loop only after the time this em. wave needs to reach it (i.e., one year in your example). So as long as you use the full Maxwell equations there cannot be any contradiction with relativity!
