Weinberg's presentation is not pedagogically ideal, because the steps are too formal, and he puts emphasis on ones that can be misleading to a student. The presentation might lead you think that the determinant of the U factors are somehow not equal to 1 naively because of some phase business, and this is categorically not true. This is what is confusing you--- the two component description makes it manifest that there is nothing with a non-unit determinant, and the four-component formulation hides this.
The part where Weinberg talks about U being pseudounitary is the misleading thing, pseudounitary or unitary, the U determinant is equal to 1, and the formal introduction of a delta function doesn't make the determinant any less 1. The quantity $e^{i\gamma_5 \alpha(x)}$ has a unit determinant because $\gamma_5$ is traceless (as Weinberg says in the middle of the calculation), so that whether you split it up into components or not, nothing changes, the measure is naively chirally invariant.
In order to define the path integral, you need to cut off the measure, and you just can't do it in a pure two-component spinor formalism consistently, not when the 2-component spinor is coupled to a gauge field. The issue is that a gauge invariant cut-off breaks the chiral symmetry. This means that you just can't make a pure two-component gauge invariant cut-off.
This only happens when you insist on a gauge invariant cutoff, otherwise you can keep the chiral current conserved (so that you keep the two spinor foralism sensible) but at the cost of making the ordinary (gauge) current not conserved. This issue is also the reason discretizing the Dirac equation is annoying--- discretization introduces a cutoff, and the cutoff must introduce other degrees of freedom in order to be consistent.
Gauge invariant cutoffs
Suppose you have a 2-spinor field which is not coupled to a gauge field, just coupled to some scalar fields. Then you can introduce a massive 2-spinor field with opposite sign kinetic term by using a Majorana mass term (in 2-spinor notation):
$$ M(\psi^\alpha\psi^\beta \epsilon_{\alpha\beta} + CC)$$
Where CC is the hermitian conjugate, with $\bar\psi$ replacing $\psi$. You can use the massive Majorana spinor to do a Pauli-villars regulator, by subtracting very massive loops from the physical light loops, leaving convergent integrals (you sometimes need several Pauli-Villars fields). Then the whole thing is fine, and the regulator is chirally invariant, it works in 2-spinor language.
What goes wrong when you couple the thing to a gauge field is that a massive Majorana spinor can't be charged. If you do the cutoff using a massive Pauli-Villars regulator, you break gauge invariance in the mass term. So you can't cut-off a single 2-component spinor with a 2 component charged spinor, not without ruining the gauge invariance and making the coupling to the vector inconsistent.
So you have some options. One superficially attractive option is to make a lattice regulator, and keep the two-component spinors. This doesn't work, because if you actually make a lattice regulator and keep the action real, you need to discretize the derivatives using a centered difference as follows:
$$ \bar\psi \sigma\cdot D \psi $$
Where D is a shift operator forward minus a shift operator backwards, using the gauge field on the link to do the shifting covariantly. The result produces a propagator with more degrees of freedom than one chiral spinor, since the lattice regulated D operator has a determinant which is zero at several places in the Brillouin zone. This is called the fermion doubling problem, and it makes a lattice regulator a nightmare.
So what you do instead is make the 2-component spinor into a four-compoent spinor, and cut off the 4-component spinor with a massive wrong-sign 4-component spinor which is charged, Pauli-Villars style. This introduces a fictitious patner to the original 2-component spinor, to make a full 4-component spinor, and coupling the whole thing to the gauge field, you only have a consistent truncation to two components only when the chiral current is conserved in any background gauge field configuration.
But now using this gauge invariant cutoff, you can compute the divergence of the axial current, and see that it isn't zero:
$$ \partial_\mu \bar\psi \gamma_5 \gamma_\mu \psi = {1\over 16\pi^2} F_{\mu\nu}F_{\lambda\sigma}\epsilon^{\mu\nu\lambda\sigma}$$
This calculation is done later in Weinberg
Fujikawa regulator
But you were specifically asking about the Fujikawa way,and when you are using Fujikawa's regulator, you takes advantage of the background gauge field to define a new set of Grassman integration variables as follows:
$$ \prod d\bar{C}_k dC_k$$
Where the $C_k$'s are a linear combination of the $\psi$'s using the k-th eigenfunction of the Euclidean Dirac operator. The point is that you can then make a gauge invariant regulator by introducing a factor of
$$e^{-k^2\over M^2}$$
into any integral over k, and this factor cuts off the short distance modes in a manifestly gauge invariant way, since it is cutting off high k modes, where k is the gauge invariant Dirac operator eigenvalue. Note that the cutoff is slightly different for every different choice of background A field, this is how the chiral current divergence ends up proportional to something involving the F tensor.
This cutoff is manifestly gauge invariant, and manifestly regulates all the Fermion integration, since it effectively cuts off the eigenvalues of the $D$-slash operator at some effective value M. So there are only a finite number of fermionic variables effectively contributing to the intermediate states of the path-integration.
Then Weinberg is evaluating the leading order correction at large M to the measure transformation using this invariant cutoff. This is why the D-slash operator appears in the cutoff that he uses, and he explains this obliquely. So when you transform the measure chirally, using the cutoff, you get the factor Weinberg gives:
$$ i\alpha(x)\gamma_5 e^{-(D\cdot\gamma)^2/M^2}\delta(x-y) $$
This Fujikawa regulator shifts the cutoff as you change the A-field, so that it is always at a given M-sized eigenvalue of $D\cdot\gamma$, and this is what allows you to calculate the anomaly when you take the cutoff to infinity. If you want to do this in 2-spinor formalism, you need to have both 2-spinors which together make the charged Fermion, so that the cutoff, either in Pauli-Villars language or in Fujikawa language, ends up being gauge invariant.
Best Answer
I think you're confusion is with notation since, unfortunately, there are two common ways to denote projected spinors. One form is to write: \begin{equation} \psi \equiv \left( \begin{array}{c} \psi _L \\ \psi _R \end{array} \right) \end{equation} In this notation $ \psi _L $ and $ \psi _R $ are two component Weyl spinors. However, a second notation is also used where \begin{equation} \psi _L \equiv P _L \psi , \quad \psi _R \equiv P _R \psi \end{equation} and $ P _{L/R } $ are the projection operators. Now $ \psi _L $ and $ \psi _R $ are four component spinors with a zero value for two the components.
In the first notation (where $ \psi _{ L/ R } $ are Weyl spinors) the Dirac term takes the form, \begin{equation} m \bar{\psi} \psi = m \left( \psi _R ^\dagger \psi _L + h.c. \right) \end{equation} and in the second (where $ \psi _{ L/R} $ are four component objects) it takes the form, \begin{equation} m \bar{\psi} \psi = m \left( \overline{ \psi _R} \psi _L + h.c. \right) \end{equation} Its just a matter of convention and the end result is the same.