Crown Princess
noun
Remember how 皇太子 is "crown prince"? Turn this crown prince into a "crown prince princess." So, this is the crown princess.
こうたいしひ
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.
皇太子妃が軍に志願するはずねぇだろうが。
The Crown Princess would never volunteer for the army.