Is there any way my son can legally work without DHS authorization?
Yes, you can apply for an Individual Taxpayer ID Number for your son at the IRS. You can visit your local IRS office or call them toll free at 1-800-TAXFORM and request a Form W-7A for Taxpayer ID Number for Pending U.S. adoptions.
Is there any way I can prove his U.S. immigration status?
To prove his status, you need a current U.S. immigration document. For example, an I-551 Lawful Permanent Resident Card or an I-766 Employment Authorization Document.