The game is beatable with any party. Even 4 white mages. Heck, people have beat it with only a single character (killing the other three and never raising them).

However, your party is far from optimal and will be more difficult. If you want a party that will have an easier time, try:

fighter, fighter, red mage, red mage = my favorite, and powerful if expensive early on.

fighter, ???, white mage, black mage = A common choice, where ??? can be any class; often a black belt or a thief. I'd go with this for a first time player, as you have more variety of classes and it is a powerful party.

But definitely take at least 1 fighter, as later in the game they are the ultimate tanks. You can win without one, but it is harder to do so.