Battlemage [looking for help]

[Submortimer]

Why use this over Eldritch knight? I'm not trying to rain on your parade, but this doesn't seem different enough from that to warrant a whole new class.

As an ability, Sword and Sorcery is a bit messy, but workable. The thing you need is something that allows you to cast ranged spells and cantrips in melee without Disadvantage; Otherwise, you're kinda limited in what spells you can use with it.

If you want to build this as a full class, Crib EK. It should have a d8 hit die, one Extra Attack (only the fighter get's multiple instances of the Extra Attack) A non fighter progression of ASI's , and essentially all of the EK's abilities.