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What I did was get my kids' Social Security numbers with their old COC, and then went back after the legal name change and just changed their name with SSA. That way they already had a SS# and were listed as US Citizens, and it was just a matter of changing their name with SSA. (For which you only need the name change documents.)

Some offices are fine with issuing a number in the new name and still listing the child as a citizen with the old COC and using the name change documents, but some offices have been known to be a pain about it.

The other alternative is to wait, go in after the name change is complete and get the number, and if they give you a hard time about the COC, just wait and go back and have your child's status changed once you have the new COC in hand.