So, I have a build idea.
4 Levels of Hellknight Signifier, 10 levels of Eldritch Knight, and either 6 levels of Wizard, Sorcerer, Witch or Arcanist. That should give me a Caster level of 19, and a BAB of 16 (so barely 4 attacks per turn) and the ability to wear Mythril Full Plate without any Arcane Failure.
As a Human with a Gladiator Background, I can get the Feats I need for both prestige classes without trouble.
I figure I can go 5 levels of Wizard or Witch, 4 Levels of Hellknight Signifier, 10 levels of Eldritch Knight, and then the Final levels of Wizard to get the missing BAB. (if i take 5 levels of Hellknight, I only have a BAB of 15 because that final level doesn’t give any).
Or, 6 Levels of Sorcerer or Arcanist, and everything is just behind a single level for a short time.
I think Human is preferred for the ability to take Martial Weapon Proficiency early, but I could swing something else.