The papers you probably should look at are the one by Hartle and Hawking establishing existence of the Majumdar-Papatetrou multi-black-hole solutions, and the one by Gibbons establishing their uniqueness.
Looking at Hartle and Hawking, if you fix a background Euclidean metric and "place" the black holes, you can place them arbitrarily close to each other as measured on the background Euclidean metric. However, this tells you pretty much nothing about any physical intuition. The problem is mainly that for a Lorentzian geometry, while the "proper time" for any two time-like related points are well-defined (modulo causality assumptions), the "distance" for any two space-like related points is not. In Riemannian geometry you can minimize over all curves connecting the two points. But in the Lorentzian case by drawing two "almost null" paths from the two points that intersect in the future, you can construct a curve in space-time connecting the two points with "length" integral as close to zero as you want. But this has nothing to do with the black hole geometry what so ever, but just a fact about reasonably nice Lorentzian manifolds.
Another way to consider distance between two objects is to compute them relative to some preferred space-like foliation. Here you run into a similar problem as before: in general you can find a foliation that is sufficiently clost to being null that the Riemannian distance along a leave of the foliation is as small as you want.
So you are down to hoping for the existence of a preferred foliation, perhaps one defined by the static time translation. Unfortunately, as you noted already in your question, the constant $t$ surfaces here do not actually intersect the event horizon (or, as you say it, it intersects the event horizon at "infinite distance"). So measured in this sense the two black holes will necessarily be "infinitely far away" from each other.
So the main difficulty, in fact, is for you to provide a definition of how you want to measure the distance between two points in Lorentzian geometry.