You can enable the Data API from CloudFormation by creating a custom resource backed lambda and enable it using any of the available SDK.
I use boto3 (python), so the lambda would have code similar as below:
import boto3
client = boto3.client('rds')
response = client.modify_db_cluster(
DBClusterIdentifier='string',
EnableHttpEndpoint=True|False
)
Obviously, you need to handle different custom resource request types and return from the lambda with success or failure. But to answer your question, this is the best possible way to set up data API via CloudFormation, for now, IMHO.
For more information about the function (Boto3):
https://boto3.amazonaws.com/v1/documentation/api/latest/reference/services/rds.html#RDS.Client.modify_db_cluster