How to transfer info from a iPhone with a dead battery to new iPhone..

If you backed up to itunes or icloud on a regular basis then click here Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

If you never backed up then there is no data to transfer.

You’ll either need a new battery to get it to boot, or you’ll need a repair shop to power it on with an external power supply and extract a backup.

Going to Apple comes with caveats. They don’t guarantee that they won’t wipe the data for any service. They may not do it if there’s other damage unless the customer agrees to get that fixed. And sometimes a “repair” involves a complete exchange in lieu of an actual repair. It’s tricky recommending Apple service when unloading data is the primary goal.