I've played three Breton battlemage (one I had to abandon because she was gltiched). My last one--which I retired after finished the main quest, building Lakehouse Manor and adopting kids but decided yesterday to bring her out of retirement--relies mostly on Alteration, Conjuration, Enchantment and Shouts. She's at 100 in those 3 schools and knows 20 shouts (13 mastered with all 3 words). She relies on shouts far more than destruction magic.
I have her in Daedric armor although her smithing at 100 and she could make slightly better dragon armor. She's the only character I've played that feels right in Daedric so I stick with that--all piece are double enchanted.
Weapon-wise she carries one of everything: an enchanted bow, an enchanted sword with a shield and an enchanted warhammer. This of curse means she can't carry a lot of loot, but she's not hurting for coin at this stage in the game.
Her main conjuration spell is Conjure Daedric Lor,, but while leveling up I preferred Conjure Flame Atronach. It's a weaker spell but I just don't like frost or storm atronachs very much. Banish daedra also comes in handy when I remember to use it. Her main alteration spells are ebonyflesh, detect life/dead and paralysis/mass paralysis.
She also sometimes carries a couple of destruction staffs, like storm wall or flame wall.
Because of her level plus ebonyflesh and enchanted armor, she doesn't really get to use restoration that much. I mostly rely on potions when her health drops unexpectedly.And i've never have got around to incorporating Turn, Ward or other non-healing restoration spells into my gameplay.
My earlier battlemages used more alchemy than my present one. My first one use destruction spells until I finally decided I just don't like the destruction spells in this game, preferring either shouts of staffs.