OK but I must point out that a primary feature of (v.del)A is its longitudinal components that are absent in vXB, And the longitudinal induction will not create the Hall effect. Still worth doing though.
Smudge
So what is for you the equivalent of vXB for A?
For me: there is no longitudinal induction in the Faraday disk, only a transverse one due to the relativistic effect of length contraction, because there is no length contraction when the observed length is colinear to the speed vector. The equivalent of vXB for A comes from the Lorentz transformation of the electromagnetic 4vector which combines A and the scalar potential.
It is named A
^{µ}. A
^{µ}=(φ/c,Ax,Ay,Az). Each coordinate of A
^{µ} are function of the spacetime position (ct,x,y,z) we are looking at. A
^{µ} is transformed into A'
^{µ} as follows:

φ/c γ γβ 0 0   0 
A'x  = γβ γ 0 0  Ax
A'y   0 0 1 0  Ay
A'z   0 0 0 1  Az
At the beginning, we have no scalar potential in the referential at rest: A
^{µ}=(0,Ax,Ay,Az). Suppose that A is only along x, so Ay=Az=0. You obtain φ/c=γβAx and A'x=γAx. We are interested here in φ/c which is the scalar potential that appears in the referential of the moving charge from the relativistic effect.
When the velocity v is different at different places in space, as is the case along the radius of a rotating Faraday disc, then φ/c=γβAx changes with the speed which is included in β=v/c and increases from the center of the disc to the rim. Therefore we obtain a potential difference along the radius, which corresponds to the electric field E=vXB that you see with the Lorentz force.
About the longitudinal induction:
in absence of a scalar potential, a longitudinal induction (say along x) means a longitudinal potential difference viewed by the moving observer (the charge) 𝝯φ/c = γAx1γAx2
≠0, or γAx(ct1,x1,0,0)γAx(ct2,x2,0,0)
≠0, i.e. that Ax has not the same value at both positions, which implies either a time dependent variation of A (classical induction), or a longitudinal spatial gradient of A along x, or a not constant speed of the observer, the charge (this last case is to be verified with GR, as the observer's referential is no more inertial).