NavList:

I believe what David McM was getting at by "it doesn't matter" is that the errors in choosing the wrong latitude to use in your calculations are smaller than the errors you'll likely have in accurately measuring your distance over bottom and maintaining a constant true azimuth.

As far as the math is concerned, the problem isn't just that "the earth isn't flat" but also "(most) rhumb lines aren't great circles."  How much the latter bit matters varies with both course and distance.

Mathematically speaking, the gold stanard, "proper" latitude to use is the latitude of the midpoint of your rhumb-line curve.  To find this, you'd find the meridional parts of both your starting and ending latitudes, find the mean between the two parts, and then use the latitude corresponding to that mean.  For a spherical earth, the merdional part of a given latitude is calculated by taking the integral of the secant of the latitude.  Modern meridional parts tables tend to refer to a non-spherical ellipsoid, usually by using a truncated series where the first term is the integral of the secant.

Strictly speaking, what I just described would be the "mid-latitude," but most texts usually end up using the term "mid-latitude" as a synonym for "mean latitude."  Using the mean of the starting and ending latitudes directly, rather than the mean of their meridional parts, tends to be "good enough" over (I believe) a few hundred miles at most inhabited latitudes.  Texts from the 19th century included tables that could be used to determine mid latitude by using both the mean and change of latittude (and assuming a spherical earth), but that seems to have gone out of fashion not long after the beginning of the 20th century.

David I