So going from the list above I'm gonna try and write a Battlemage spell for each level. These would be on the Battlemage list, Wizard list (to not leave out EK and Bladesinger), and maybe divided amongst some other spell lists depending on flavor. All spells have no Verbal component, a Somatic component (technique), and a Material component (a weapon). These spells will be based and named after 4e Swordmage powers just so I can focus on balancing over creativity.
Please critique as you see fit.
Arcane Lance
Evocation Cantrip
Casting Time: 1 action
Range: Self (10-foot radius)
Components: S, M (a melee weapon worth at least 1 sp)
Duration: 1 round
You brandish the weapon used in the spell’s casting and thrust it towards an enemy. The target must make a Dexterity saving throw or suffer the weapon attack’s normal effects and damage and have it's speed reduced by 10 feet until the end of your next turn.
If you take the Dash action on your turn you may cast this spell as a bonus action.
At Higher Levels. At 5th level, the melee attack deals an extra 1d8 thunder damage to the target on a hit, and an additional 1d8 thunder damage at 11th level (2d8), and 17th level (3d8).
Blazing Pursuit
1st-Level Evocation
Casting Time: 1 bonus action
Range: 30 feet
Components: S, M (a melee weapon worth at least 1 sp)
Duration: Concentration, up to 1 hour
You teleport up to 30 feet towards your target and place a blazing mark upon them. While marked in this way your attacks against that creature deal an additional 1d6 fire damage and you may teleport up to 30 feet as a bonus action on each of your turns as long as you end the teleport closer to your target than you started.
At Higher Levels. When you cast this spell using a spell slot of 4th level or higher it no longer requires concentration.
Blades of Fiery Wrath
2nd-Level Conjuration
Casting Time: 1 reaction, which you take when you miss with a melee weapon attack
Range: 5 feet
Components: S, M (a melee weapon worth at least 1 sp)
Duration: Instantaneous
The ambient magic of your unsuccessful attack manifests as countless blades of fire that lash out at nearby enemies. When you miss with a melee weapon attack as a reaction you may cause all creatures within range to make a Dexterity saving throw, taking 3d8 fire damage on a failed save or half as much on a successful save.
At Higher Levels. When you cast this spell using a spell slot of 3rd level or higher the damage increases by 1d8 for every spell level above 2nd.
Enervating Slash
3rd-Level Necromancy
Casting Time: 1 bonus action
Range: Self
Components: S, M (a melee weapon worth at least 1 sp)
Duration: Concentration, up to 1 minute
The next time you hit a creature with a melee weapon attack during this spell’s duration, your weapon turns pitch black as you strike, dealing an additional 3d10 necrotic damage. Additionally, the target must succeed on a Constitution saving throw or deal half damage with any of its attacks. The creature may repeat the saving throw at the end of each of its turns, ending the effect on a success.
At Higher Levels. When you cast this spell using a spell slot of 3rd level or higher the damage increases by 1d10 for every spell level above 3rd.
Silverlight Strike
4th-Level Evocation
Casting Time: 1 bonus action
Range: Self
Components: S, M (a melee weapon worth at least 1 sp)
Duration: Concentration, up to 1 minute
The next time you hit a creature with a melee weapon attack during this spell’s duration, your opponent is surrounded by blinding lights that obscure your allies. All creatures except for you are considered invisible to the target for the duration. The target may make a Constitution saving throw at the end of each of its turns, ending the effect on itself on a success.
Planar Shock
5th-Level Conjuration
Casting Time: 1 bonus action
Range: Self
Components: S, M (a melee weapon worth at least 1 sp)
Duration: Concentration, up to 1 minute
The next time you hit a creature with a melee weapon attack during this spell’s duration, you send them flying through an otherworldly space. The target takes an additional 5d10 psychic damage and must make a Wisdom saving throw or be teleported up to 90 feet to a destination of your choice and be stunned. The creature may repeat the saving throw at the end of each of its turns, ending the stunned condition on a successful save.