Should I erase my iPad Air before sending it off for a battery replacement?

Personally, I would create an iCloud backup and factory reset the device before giving it to anyone.

How to back up your iPhone, iPad, and iPod touch - Apple Support

How to factory reset your iPhone, iPad, or iPod touch - Apple Support

For practical reasons, in lieu of replacing the battery and returning your own iPad, Apple and its repair agents usually exchange the iPad for a renewed/refurbished iPad of the same model and specification. The cost of the exchange is the same as would otherwise be charged for the battery replacement.

Before returning your iPad for battery replacement or exchange, ensure that you have a backup of your data - as any locally stored data will be lost. Follow this process to properly prepare your iPad for service:

Prepare your iPhone or iPad for service - Apple Support

When you receive the your iPad back from service, set-up the iPad - restoring your data and settings using your pre-existing backup. If you have an iCloud backup, you will be prompted during the initial setup process - or you can manually restore your data from iCloud or your Mac computer or Windows PC after completing the setup process:

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support