"The issue is if, under legal guardianship, he qualifies as a Qualifying Child or any other classification that allows the EIP and Child Tax Credit to go to my client."

Now that you see how to determine this is a Dependent, you can use the EIP links I gave to use that info for EIP purposes. The same is true for CTC.

"I did not see any language in your links discussing a child who is under "guardianship"."

Guardianship is not the IRS terminology.