First of all, if the child is really this friend's biological child, and your friend met the requirements under US law to transmit citizenship to a child born abroad, the child is already a US citizen under US law, from birth, and thus cannot "obtain" it. The question is only whether they will be able to prove this citizenship.
Acquisition of US citizenship by birth abroad from a parent requires blood relationship*. A child who is born while the mother is married (as in this case) is generally presumed to be the biological child of that couple, but this is not determinative, and the consulate has discretion to investigate and request additional evidence if they suspect the child might not be the biological child.
d. Children born in wedlock are generally presumed to be the issue of
that marriage. This presumption is not determinative in citizenship
cases, however, because an actual biological relationship to a U.S.
citizen parent is required. If doubt arises that the U.S. citizen
"parent" is biologically related to the child, the consular officer is
expected to investigate carefully. Circumstances that might give rise
to such a doubt include, but are not limited to:
(1) Conception or birth of a child when either of the alleged
biological parents was married to another person during the relevant
time period;
(2) Naming on the birth certificate, as father and/or mother,
person(s) other than the alleged biological parents; and
(3) Evidence or indications that the child was conceived at a time
when the alleged father had no physical access to the mother.
(4) If the child was conceived or born when the mother was married to
someone other than the man claiming paternity, a statement from the
man to whom the mother was married disavowing paternity, a divorce or
custody decree mentioning certain of her children but omitting or
specifically excluding the child in question, or credible statements
from neighbors or friends having knowledge of the circumstances
leading up to the birth may be required as evidence bearing on actual
natural paternity.
(5) The child was born through surrogacy or other forms of assisted
reproductive technology. (8 FAM 304.3 provides guidance
about acquisition of U.S. citizenship by birth abroad and assisted
reproductive technology.)
e. In such cases, it is within the consular officer's discretion to
request additional evidence pursuant to 22 CFR 51.45.
When there is doubt about the paternity, some things the consulate can do include obtaining records of the periods of time the father had physical access to the mother, interviewing the mother and father separately about when and where the child was conceived, interviewing neighbors and friends, and, if the parents want to pursue the claim even if the facts don't seem to support it, they can advise the parents to do a DNA test.
We don't know whether the consulate will find the other evidence of paternity for this child sufficient, or whether they will deem it necessary for your friend to do a DNA test. If the consulate deems a DNA test necessary and your friend doesn't do it, they can't prove the US citizenship of the child. In that case, if your friend and the child are moving to the US, maybe your friend can petition the child to immigrate (i.e. to become a US permanent resident); this doesn't require a blood relationship -- stepparents can petition as long as the child was under 18 when the marriage occurred. But I am not sure whether the consulate will issue the immigrant visa in such a case where the child's US citizenship isn't proved only due to the parent's refusal to do DNA test.
*Update: Recent case law has thrown some doubt onto whether a blood relationship is really required. The 2nd Circuit court of appeals (which covers Connecticut, New York, and Vermont) ruled in the case Jaen v. Sessions (2018) that someone who was born to a married couple but who was not the biological child of his legal father nevertheless did acquire US citizenship from his legal father at birth. I don't know if this will be appealed and whether it will apply to other circuits. If blood relationship isn't required, then perhaps a DNA test won't be necessary.
