Is it permissible to spend a portion of the selling price of rabbits designated for Allah on the mother and her young, or must the entire selling price be given for the sake of Allah, with their expenses covered from a personal account?

This falls under the category of ongoing charity (Sadaqah (definition) Jariyah) related to Waqf (endowment). The endowment of an animal is permissible according to the majority of scholars. There is no harm in spending part of the price of the young of the endowment for the benefit of the rest of the endowment. The expenses of the endowment are to be covered from its revenue unless stipulated otherwise, and it is permissible to spend on the endowed mother from the endowment's revenue.