But an axiom that requires demonstrations is not an axiom at all.
Every deductive system is based on indemonstrable axioms and deduced theorems (with transformation rules). If a sentence in this system needs to be demonstrated is not an axiom, is a theorem.
A moral principle is the basic postulate of a moral system. If the whole system must be demonstrated you can show the external principles that demonstrate the basic moral principles. Then, you are saying that moral is not autonomous. It depends from religion, science or astrology.
But you have an unsolvable logic problem: in logic, the connectors of the conclusion must be included in the premises. You cannot conclude a universal sentence from particular sentences, for example. And you cannot conclude a sentence with “ought to” from sentences of the form “is”. But science is a system of descriptive/explicative sentences, that is to say, with the form “is/are”. Therefore, you cannot deduce moral principles from scientific laws. (Hume’s guillotine).
Note that the impediment is of different level than Fermat’s theorem. This one neither violates the logic rules nor mathematics axioms. It was only a problem of find the solution inside the mathematic system. Hume’s guillotine is unsolvable because raises a contradiction with basic rules of logic.