Is it permissible for a veiled Muslim woman to study fashion design in a mixed course with a small number of male students, and her instructor is female, given that a more expensive alternative is available, and is learning this art necessary and beneficial for a Muslim woman?

There is no harm in participating in fashion design courses, provided that Islamic legal controls are adhered to, such as avoiding temptation (fitna), lowering the gaze, and maintaining modesty. Mixing (of genders) is forbidden if it leads to prohibited acts, such as being alone with a non-mahram (definition) woman (khalwa (definition)), looking with lust, and not observing modesty. As for learning fashion design, it is beneficial as a means of earning a livelihood when needed, and it is among the luxuries, not the necessities.