In this paper Bell derives his famous inequality using the assumtions of locality and realism. In order to understand how the locality assumption affects the derivation of the inequality, and why it is needed for the equality, I have attempted to re-derive the inequality, first assuming locality and then a second time assuming nonlocality, to see what the difference is. However, my derivations seem to say that there is no difference, which implies that nonlocality cannot be concluded from a Bell test, which is wrong (or some other, smarter, mathematician would have pointed it out by now). Where am I making my mistake(s)? Note: I know there are other similar questions regarding nonlocality in the CHSH inequality. I have read them and I don't see their application to this (the original) form of Bell's inequality (they use different mathematical formalism and expression which I do not see appear in Bell's original derivation).
The system is a pair of entangled particles. Let $A = \pm 1$ be the result of Alice's measurement of one particle's spin, and let $B = \pm 1$ be the result of Bob's measurement of the other's. Let $\mathbf{\alpha}$ and $\mathbf{\beta}$ be unit vectors representing Alice and Bob's measurement directions respectively. Let $\lambda$ represent a set of any number of hidden variables and $\rho = \rho(\lambda)$ the normalized probability distribution of $\lambda$.
As far as I can tell, the locality assumption amounts to assuming that $A = A(\mathbf{\alpha}, \lambda) \neq A(\mathbf{\alpha}, \mathbf{\beta}, \lambda)$, or that $A$ is independent of $\mathbf{\beta}$, and likewise for $B$ and $\mathbf{\alpha}$ (this may be my mistake if there is more to it than this).
Local derivation: $A(\mathbf{\alpha}, \lambda) = \pm 1$, $B(\mathbf{\beta}, \lambda) = \pm 1$. The expectation value of $AB$ is
\begin{equation} P(\mathbf{\alpha}, \mathbf{\beta}) = \int \rho A(\mathbf{\alpha}, \lambda) B(\mathbf{\beta}, \lambda)\, d \lambda. \end{equation}
For a given measurement direction $\mathbf{a}$,
\begin{equation} P(\mathbf{a}, \mathbf{a}) = \int \rho A(\mathbf{a}, \lambda) B(\mathbf{a}, \lambda)\, d \lambda = -1 \implies A(\mathbf{a}, \lambda) = -B(\mathbf{a}, \lambda). \end{equation}
$P(\mathbf{a}, \mathbf{a}) = -1$ implies that the particles are anticorrelated, and so by rewriting the expectation value of $A B$ as
\begin{equation} P(\mathbf{\alpha}, \mathbf{\beta}) = -\int \rho A(\mathbf{\alpha}, \lambda) A(\mathbf{\beta}, \lambda)\, d \lambda \tag{1} \end{equation}
(in other words, by assuming $A(\mathbf{\alpha}, \lambda) = -B(\mathbf{\beta}, \lambda)$ is always valid) we mathematically represent the assumption that the state of our two-particle system is restricted to a maximally anticorrelated state ($| \Psi^\pm \rangle$). Using this last expression, we get (for some unit vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$)
\begin{align} P(\mathbf{a}, \mathbf{b}) - P(\mathbf{a}, \mathbf{c}) =& -\int \rho \Big( A(\mathbf{a}, \lambda)A(\mathbf{b}, \lambda) - A(\mathbf{a}, \lambda) A(\mathbf{c}, \lambda) \Big) d\lambda \\ =& -\int \rho A(\mathbf{a}, \lambda)A(\mathbf{b}, \lambda) \Big( 1 - \frac{A(\mathbf{a}, \lambda) A(\mathbf{c}, \lambda)}{A(\mathbf{a}, \lambda)A(\mathbf{b}, \lambda)} \Big) d\lambda \\ =& \int \rho A(\mathbf{a}, \lambda)A(\mathbf{b}, \lambda) \Big( A(\mathbf{b}, \lambda) A(\mathbf{c}, \lambda) - 1 \Big) d\lambda, \end{align}
\begin{equation} |P(\mathbf{a}, \mathbf{b}) - P(\mathbf{a}, \mathbf{c})| \leq \int \rho \Big( 1 - A(\mathbf{b}, \lambda)A(\mathbf{c}, \lambda) \Big) d\lambda = 1 - P(\mathbf{b}, \mathbf{c}), \end{equation}
\begin{equation} |P(\mathbf{a}, \mathbf{b}) - P(\mathbf{a}, \mathbf{c})| + P(\mathbf{b}, \mathbf{c}) \leq 1. \end{equation}
Nonlocal derivation: $A(\mathbf{\alpha}, \mathbf{\beta}, \lambda) = \pm 1$, $B(\mathbf{\beta}, \mathbf{\alpha}, \lambda) = \pm 1$. The expectation value of $AB$ is
\begin{equation} P(\mathbf{\alpha}, \mathbf{\beta}) = \int \rho A(\mathbf{\alpha}, \mathbf{\beta}, \lambda) B(\mathbf{\beta}, \mathbf{\alpha}, \lambda)\, d\lambda. \end{equation} \begin{equation} P(\mathbf{a}, \mathbf{a}) = \int \rho A(\mathbf{a}, \mathbf{a}, \lambda) B(\mathbf{a}, \mathbf{a}, \lambda)\, d\lambda = -1 \implies A(\mathbf{a}, \mathbf{a}, \lambda) = -B(\mathbf{a}, \mathbf{a}, \lambda), \end{equation} \begin{equation} P(\mathbf{\alpha}, \mathbf{\beta}) = -\int \rho A(\mathbf{\alpha}, \mathbf{\beta}, \lambda) A(\mathbf{\beta}, \mathbf{\alpha}, \lambda)\, d\lambda, \tag{2} \end{equation} \begin{align} P(\mathbf{a}, \mathbf{b}) - P(\mathbf{a}, \mathbf{c}) =& -\int \rho \Big( A(\mathbf{a}, \mathbf{b}, \lambda) A(\mathbf{b}, \mathbf{a}, \lambda) - A(\mathbf{a}, \mathbf{c}, \lambda) A(\mathbf{c}, \mathbf{a}, \lambda) \Big) d\lambda \\ =& -\int \rho A(\mathbf{a}, \mathbf{b}, \lambda) A(\mathbf{b}, \mathbf{a}, \lambda) \Big( 1 - \frac{A(\mathbf{a}, \mathbf{c}, \lambda) A(\mathbf{c}, \mathbf{a}, \lambda)}{A(\mathbf{a}, \mathbf{b}, \lambda) A(\mathbf{b}, \mathbf{a}, \lambda)} \Big) d\lambda, \end{align} \begin{equation} |P(\mathbf{a}, \mathbf{b}) - P(\mathbf{a}, \mathbf{c})| \leq 1 - \int \rho \big( A(\mathbf{a}, \mathbf{c}, \lambda) A(\mathbf{c}, \mathbf{a}, \lambda) A(\mathbf{a}, \mathbf{b}, \lambda) A(\mathbf{b}, \mathbf{a}, \lambda) \big) d\lambda, \end{equation} \begin{equation} |P(\mathbf{a}, \mathbf{b}) - P(\mathbf{a}, \mathbf{c})| + \int \rho \big( A(\mathbf{a}, \mathbf{c}, \lambda) A(\mathbf{c}, \mathbf{a}, \lambda) A(\mathbf{a}, \mathbf{b}, \lambda) A(\mathbf{b}, \mathbf{a}, \lambda) \big) d\lambda \leq 1. \end{equation}
Question: My result is of the same form as Bell's, but I cannot simplify the third term on the left to $P(\mathbf{b}, \mathbf{c})$, so the third term retains its nonlocal dependence on $A$'s second argument. Despite this, both $\int \rho \big( A(\mathbf{a}, \mathbf{c}, \lambda) A(\mathbf{c}, \mathbf{a}, \lambda) A(\mathbf{a}, \mathbf{b}, \lambda) A(\mathbf{b}, \mathbf{a}, \lambda) \big) d\lambda$ and $P(\mathbf{b}, \mathbf{c})$ are restricted to the range $-1 \leq x \leq 1$, so both inequalities should lead to the same experimental conclusions regarding local realism. So what difference does the locality assumption make? What assumption am I misrepresenting? Or what other mistake am I making?