I think it is great that you desire for your family to become permanent residents of the USA by obtaining a green card. Following the links provided on that webpage (While your green card application is pending, USCIS Processing Times, Historical Average Processing Times) may provide you with information to help manage your expectations about the length of time remaining until you receive a decision on the green card. From prior posts it appears that your child is about 8 years old, so there is not a time crunch to apply to DYS... there are about 8 more years to apply.

Is potential application to the DYS program your sole reason for having your child tested?

It seems a bit unusual for a member of the forum for more than two years not to have seen the information on the How to Apply - DYS Eligibility Requirements webpage prior to scheduling an appointment for assessment. The same webpage states:
Originally Posted by How to Apply
For more information about this process, please visit the Frequently Asked Questions about the Young Scholars program and application process.

If your question is not answered in the FAQs, please email ysapplications@davidsongifted.org. Emails will be answered within three to five business days.