I think it would be difficult to fit Red Mage into the current system (or even pre 1.19) with cross class abilities being the way they are (limited to no duplicates).

You might have a Fencer, with a Red Mage job who has Conjurer and Thaumaturge as their cross class abilities.

Red Mage, being a jack-of-all-trades type, with many fighter weapons and armor available, using an array of white and black magic spells, but not specializing in any one area.

Although I'm sure the lore could be adapted in FFXIV. Where the red mage is a subset of fencers who utilize magic to enhance their effectiveness. Be that Red Lotus for a flame sword attack or an enfeebling attack like Swarmstrike (weapon skill that afflicts poison).