Is it permissible for an unofficial foreign mosque imam to take money from an official muezzin in exchange for continuing to lead prayers, out of fear that no other imam would be available?

There is no harm in taking money for dedicating oneself to leading people in prayer (Imamah). Taking a wage for leading prayers is permissible, especially when there is a need for it, and this is what Sheikh Taqi al-Din chose. If you wish to refrain from taking money, then do so and seek out someone from the mosque community to lead them in prayer. Consider which of the two benefits is more significant to achieve, and which of the two harms is greater to avert.