Here's something to think about, at least (not elegant, not a solution, but
perhaps a guide).
Assume a magnetostatic situation (by which I mean constant current and no
changing electric fields). Further assume that the current density is in
the positive z direction. The current-carrying wire may be cylindrical and
have arbitrary diameter.
Consider the differential form of Ampere's law in MKS units (curl B = mu_0
J). At all points in space (both inside and outside the wire), J_x = 0
(since J_z = J). Therefore (curl B)_x = 0. But (curl B)_x = (DB_z/Dy -
DB_y/Dz), where the capital D's indicate partial derivatives. Note that I
am using rectangular coordinates even though this is a cylindrical problem,
partly because I forget how to write the curl in cylindrical coordinates.
Due to symmetry, B_y (which is part of the azimuthal field component) has
no z-dependence (otherwise we would be able to tell where we are along the
length of the infinite wire), so DB_y/Dz must be zero. Therefore, DB_z/Dy
must also be zero, which constrains B_z to be (at best) a constant function
of y (of course, what we would really like to show is that the constant
must be equal to zero).
A similar argument (using the y-component of Ampere's law) can be made to
show that B_z is also (at best) a constant function of x.
Since B_z also cannot be other than a constant function of z (if it were
non-constant, we could use it to determine our position along the wire), we
conclude that B_z is, at best, constant throughout space.
I'll leave it as an exercise for you to figure out why a non-zero constant
value of B_z would be unphysical.
There is undoubtedly a more beautiful way to get this, but since this is a
first crack based on memory alone (no textbooks were consulted), I hope
you'll excuse the inelegance of the approach.
P.S. I also have a vague memory (memorism?) of Jackson's (in Classical
Electrodynamics) deriving the Biot-Savart law from the Maxwell equations.
(I imagine that the derivation can't be trivial; otherwise, intro-level
texts would do it.) If this memory is correct, then allowing use of the
Maxwell equations while eschewing Biot-Savart seems kind of an odd thing to
do. I'd look it up, but I'm on spring break and my books are at school.
Maybe a list member can check this.