What do you do if your kid is listed as an "adult" and there is no "ask to buy" option listed?

The only thing that determines an adult from a child is the age set for the Apple ID. There is no actual setting beyond the age for this.

Update the Apple ID and change the birth date to something correct, then Ask To Buy should be able to be turned on.

link-> Update the birth date associated with your Apple ID - Apple Support

Ask to Buy applies to children. Not adults.

Approve what kids buy with Ask to Buy - Apple Support

Per Apple, “Don't want to pay for purchases of a specific adult family member? Remove that person from your family group or turn off purchase sharing for everyone.“