Yes, but that is a rather general problem. Best you can do is to look at the so called Lyman-alpha forest, which gives you essentially the mass distribution along the path. And this gives you a good estimate for the deflection.

The mass distribution isn't necessarily a good proxy for the deflection. Lyman alpha lines are from absorption by hydrogen gas clouds; the average effect of those on the path of light is likely to be close to zero, because the clouds are diffuse and more or less homogeneous (i.e., their average density on large length scales is pretty much the same everywhere). The "Tardis region" model requires large deflections of light at the boundaries between the "Tardis" regions and regions of normal density, which will not average out since there aren't enough such boundaries between us and the distant supernovas that are our main line of evidence for the accelerating expansion of the universe.

I did not read the tardis paper, but the mass distribution gives you at least a upper limit, in the sense that if there is not much mass in one direction, then there is also not much deflection. At least in more or less normal models.

In more or less normal models, yes; but as I read the Tardis paper, the whole point is that it is not a more or less normal model.