I recently bought a Venu 2 watch. How can I add Badminton as an activity?

Badminton is only available as an activity profile on the Venu 3 - Activity Profile Options On My Wellness Watch

You can edit the activity after uploading to Garmin Connect and change it to badminton, but it won't be the same as tracking the activity from the watch as badminton.

You can also add it as a custom activity, but again, will not be the same as having it natively on the watch because you will be copying another activity  - Venu 2 Series Owner's Manual - Adding a Custom Activity

Otherwise, you can try using one of the Connect IQ apps, such as Badminton | Garmin Connect IQ.

If you check the app store, there are apps that will allow you to record an activity, like badminton, that the watch doesn't support but Garmin does.  I use Any Sport PRO and it works great.