what class beats a battle mage?

As always, it depends a bit on the specific champion. Not every burst mages gets countered by assassins, for example, like Lissandra.

But in general I would guess that artillery mages do well against them. Battle mages tend to have shorter ranges than other mage classes. An artillery mage specialises in long range combat, wittling the battle mage down over time. And most of the sustain from a battle mage (if they got any) mostly comes from combat with champions. But if they can't reach the artillery mage, then they can't use it.

Battlemages as a class focus on sustained damage. Anything that builds and the deliberately anti-Magic champions ( ) specifically do strongly against the kind of damage battlemages output.

They also have the lowest mobility in the game and no escape options outside of pool, so any of the hypermobile recent releases can easily outplay many of them

And finally, anything with good waveclear/poke can keep battlemages under tower while they roam and make plays

Any assassin ever.

battle mages?

define please.

Assassins still long as their burst is on the high end.

Reason being that battle mages tend to have extra HP cause of their typical builds and extra defenses through shielding and sustain.