In a Bell local hidden variable model, is marginal measurement independence sufficient, or must the joint distribution also be setting-independent?

I've been thinking about Bell's theorem and got confused about something..

Suppose a source emits particle pairs and each particle carries a hidden variable. So i can denote Alice's effective hidden variable as λ_A (which may depend on Alice's setting a) and Bob's as λ_B (which may depend on Bob's setting b).

Now suppose the following is true
1.The marginal distribution of λ_A alone does not depend on the settings a or b.
2.The marginal distribution of λ_B alone does not depend on the settings a or b.
3.But the joint distribution P(λ_A, λ_B) does depend on both settings through their relative angle θ = θ(a, b).

Now my question is -
Does such a model satisfy Bell's locality condition? Or does Bell's theorem require that the joint distribution P(λ_A, λ_B) also be independent of the measurement settings?

Intuitively I suspect this violates locality because Alice's outcome ultimately depends on Bob's setting through the joint structure — but I want to understand this precisely in terms of Bell's factorization condition.