I recommend using the same character. Circumventing the mythology cap and being able to do more coil groups are the only advantages to having a new character. By using the same character, you have the armory bonus to help level things up, easier access to cross-class abilities (Unless you plan to do a LOT of leveling), and can work on gearing up other classes while doing other content (For example, if you see a garuda sword drop, but you're on your WHM, or using philosophy tomestones you get while farming myth to get darklight), as well as having the ability to share gear such as accessories, or just about any tanking/healing/casting gear.

Another thing to keep in mind: There's not much need to rush to get everything on everything ASAP. If you're only working on one job per character, it shouldn't take THAT long to get all 90+ gear. Then what?