According to the Player's Handbook Ch. 9, "force" is a damage type, and "damage types have no rules of their own."

The Awakened Spellbook feature of the Order of Scribes Wizard states that under certain circumstances when casting a spell, "you can temporarily replace its damage type with a type that appears in another spell in your spellbook".

The Mage Armor spell states, "You touch a willing creature who isn't wearing armor, and a protective magical force surrounds it until the spell ends. [...]" (emphasis mine).

It is well established that if the only spells in your spellbook were Burning Hands and Magic Missile, you could cause Burning Hands to deal force damage. But if the only spells in your Wizard's spellbook are Burning Hands and Mage Armor, could you still cause Burning Hands to deal force damage because "force" "appears in" Mage Armor?

I'm asking from a purely R.A.W. (Rules As Written) perspective—I'm fully aware that many DMs would rule against it, but I'd like to know whether doing so would be considered a house rule or an interpretation of the rules as written.