He can't even get the exemption without the ITIN. What you can do however is file for an ITIN with the W-7 attached to the return claiming the child and if qualified CTC. Does the child live in the US?
Print out the return and put the CTC in by hand. You cannot e-file these returns and apparently you can't override the software - and no the IRS will not correct it for you.
Also for the child's SSN write in applied for.
Read the W-7 instructions carefully. You will have to supply a passport or original birth certificate.
If your client does not want to send these documents in, you can get copies approved by an Acceptance Agent. Look in irs.gov for a list of these in your area.
The last time I checked, these returns are mailed to Austin and not to your regular filing address.