"In the case of Pakistan, the UN Medium Fertility Case projects that the population will rise from about 165 million today to 344 million by 2050 - a rise of 110%."

"The effect of combining such a drop with the projected rise in population is obvious: a drop of 66% in average per capita GDP. "

Of course, there is one thing you seem not to have considered - one or both of these projected trends might impact the other. For example, medium sized families that are already struggling for cash might decide not to become large families. Alternatively or additionally, poverty induced by the second trend could lead to earlier death and so reduce the population rise. It seems to me the UN population projection shows what would happen if the resource base existed to support it, while the peak oil trend simply says that the resource base will not exist. As such, I don't think mapping it out 40 years and saying "it'll be a catastrophe" is useful - because before it becomes so it will be a trend with it's own negative feedback.
Of course I am assuming that the people of Pakistan would, on the whole, rather have fewer children than watch some of them starve. If they choose to do things differently, well, that really is their business.

The eventual thrust of the argument I'm starting to make here is that these trends are in fact antithetical. This initial positioning simply outlines the influences using some well-accepted (in the case of the UN projection), academic (in the case of Kummel and Ayres) and speculative (in the case of my energy analysis) trends and relationships.

There is obviously going to be feedback. The question is what the feedbacks paths are going to be. If an outcome is unsupportable, as this one may well be, there are several ways events could unfold. Energy inputs (and their derivatives, food and GDP) might be increased above the projection by redirecting discretionary expenditures to the energy sector. Population might fall below the projections, either voluntarily through fertility control or involuntarily through mortality increase. Efficiencies may be introduced to make better use of the existing energy. Some combination of all these (increased energy, decreased fertility, increased mortality and increased efficiency) is probable, but the relative contributions of each are quite speculative at this point. Which factors one believes will dominate depends at least as much on the mindset of the analyst as on the data.

What does not seem speculative is that in a country where energy availability is declining as population continues to rise, and the combination of those two slopes exceeds the rate at which efficiencies (or foreign aid) can be introduced, eventually something has to give.