Is there something wrong with arguing that since the problem does not specify an orientation for the sphere, rotating the sphere should not affect the resulting distribution arrived at by the chosen limiting process?
One one hand, we have a pre-theoretic, intuitive understanding of probability. On the other, we have Kolomogorov's formal axiomatization of probability.
The principle of indifference belongs to our intuitive understanding of probability. We feel that any formalization of probability should respect it. However, as you note, our formal theory of probability does not always do this, and the the Borel-Komogorov paradox is one of the cases where it doesn't.
So, here's what I think you're really asking: How do we resolve the conflict between this attractive intuitive principle and our modern measure-theoretic theory of probability?
One could side with our formal theory, as the other answer and the commenters do. They claim that, if you choose the limit to the equator in the Borel-Kolmogorov paradox in a certain way, the principle of indifference does not hold, and our intuitions are incorrect.
I find this unsatisfactory. I believe that if our formal theory does not capture this basic and obviously true intuition, then it is deficient. We should seek to modify the theory, not reject this basic principle.
Alan Hájek, a philosopher of probability, has taken this position, and he argues convincingly for it in this article. A longer article by him on conditional probability can be found here, where he also discusses some classic problems like the two envelopes paradox.
I don't see the point of the "principle of indifference". The Wikipedia article's answer is better: "Probabilities may not be well defined if the mechanism or method that produces the random variable is not clearly defined." In other words, without even restricting ourselves to questions of probability, "An ambiguously-posed question does not have a single unambiguous answer."