A transfer of the case is typically to speed up the processing of your case and has nothing to do with the child's eligibility for a social security number. With an EAD your child should be able to get a social security number, but it would be restricted and say not valid without employment authorization. Most likely they are telling you to wait because when the child gets the green card they can issue a social security number without restriction.
Applications are often transferred between offices. This is not a problem. You can apply for a SS card for your child. The A# will be on the EAD work permit.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.
Transfers are routine. You may apply for SS# for your child as you will probably need it for tax filing considering there may be a slight delay due to the transfer.
This response is general in nature and cannot be construed as legal advice, given that not enough facts are known. Any comments offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship. You are encouraged to seek independent and private counseling for a complete review of your case.