I suspect that we're only talking to ourselves at this point, so perhaps it's not worth a lot more time on this conversation. But a few notes:

Regarding the infinite plate limit: Locally, it is absolutely true that as you increase the size of the plates (relative to the distance between them) the fields must approach the result for an infinite plate. But as soon as you want to consider global questions like the behavior of the potential at infinity, there is a true qualitative difference between a large-but-finite capacitor and an infinite one. Any finite charge distribution will result in a potential of zero at infinity (or rather, will allow you to consistently make that choice), but an infinite charge distribution will not in general allow that. For the case of an infinite parallel plate capacitor, if you set the potential to zero out to infinity on one side, it will be non-zero (and constant) out to infinity on the other. One upshot of this is that it's misleading at best to use the infinite plate approximation for any sort of scattering analysis (like what we're discussing here).

Regarding multipole expansions: I love multipole analysis, it's a beautiful way to study charge distributions. That said, I think it's massive overkill to actually use it in this case, since at least for circular plates along the axis that process is far more complicated than just using the exact fields themselves.

Regarding momentum transfer: You've claimed that "It is the functional form of the potential out to infinity which determines how much momentum the ions transfer back to the device". I agree that if you want to know the rate of momentum transfer at any given point, you need to know the exact functional form of the potential (and/or the field). But if we're only interested in the total momentum transferred in the interaction, we don't need to know the details of where and when the momentum transfer happened: all we need to know is the net result. So in particular, if your ion starts far in front of the device with (relative) velocity v0 in the -x direction, and we can show from energy conservation that it ends far behind the device with the same velocity v0 in the -x direction, that's sufficient to prove that the ion has experienced no net change in velocity and thus that no net momentum transfer has occurred (even though there may have been substantial transfers along the way).

With that in mind, if we specialize to the simple case of an ion moving along the axis of symmetry of two circular plates, we can find exactly this result explicitly from an energy calculation. So to heck with it, let's just do this. (And apologies if this is repetition, but I don't think I was able to get your link in that other thread to work, or perhaps I tried to follow the wrong one.) Along the axis of a uniformly charged circular disk of radius R,

E_x(disk) = K * (1-|x|/sqrt(x^2+R^2)) * sign(x),

where K is a constant (K=sigma/(2 epsilon_0) for surface charge density sigma). Assembling two of these with opposite charges a distance D apart (with the + plate at x=0 and the - plate at x=-D) gives for x>0:

E_x(x>0) = K * [(x+D)/sqrt((x+D)^2 + R^2) - x/sqrt(x^2 + R^2)]

This has the properties you'd expect: for example, it approaches D/R->0 as R approaches infinity, matching the infinite plate limit. (The expression's clearly different between the plates.) You can integrate this (or rather, its negative) to get the electric potential:

phi(x>=0) = K * [sqrt(x^2 + R^2) - sqrt((x+D)^2 + R^2) + D]

The constant +D was chosen to maintain the convention phi=0 at infinity. But then we find that the potential at x=0 is

phi(x=0) = K * [R - sqrt(D^2 + R^2) + D],

or approximately K * D in the limit of large R (and this actually decreases as R gets smaller). [This is consistent: by symmetry, phi=0 at the midpoint between the plates (x=-D/2), and in the large R limit the electric field inside is a constant sigma/epsilon_0 = 2K, so 2K * D/2 = K * D.] No matter how large you make the plates, the total change in the incoming ion's energy and thus the total effect of the exterior field on the ion's speed will not approach zero, even though the magnitude of the field at any given point does approach zero as R grows (as it must, to match the infinite plate limit). And no matter how large you make the plates, the total effect of the field on the speed of an ion that moves from infinity to the midpoint of the plates will be zero (and similarly on the negative/outgoing side). You're welcome to solve equations of motion to find the exact functional form of x(t) for this particle, but that's a lot of work, and energy conservation guarantees that there will be no net change in speed.

Finally, regarding plasma screening: My comment about positive ions being strongly repelled outside of the positive plate was perhaps unclear, and certainly I'd welcome corrections if I'm misunderstanding screening somehow. I was speaking specifically of ions that came closer to the positive plate than the length scale over which the screening brings the outside field to zero. The screening charges can't change the electric potential at the position of the plate itself (unless you allow them to discharge the whole thing), so rather than falling off to zero very slowly over a long distance as in the unscreened case (giving rise to a small but wide electric field as the gradient), the potential must fall to zero very rapidly over the short screening distance (giving rise to a large but short-ranged electric field). Either way, the height of the potential energy barrier at the plate (relative to infinity) can't change. So again, in the screened case I would think that incoming positive ions would feel essentially zero field for most of their approach but would then feel a very strong repulsion in the narrow region immediately before the plate.