What legal ways can one prove fatherhood if the mother refuses to give
access to the child?

DNA from the child and from the putative father will be required.
Your problem devolves to obtaining the child's DNA legally, but without access.
Your most likely source would be something discarded by the child into
the public domain.

Greetings!

 I agree with xarqi inasmuch it is concerned with child-putative father DNA test.

 You may establish paternity/fatherhood by establishment by the court
after filing a legal action (court case), wherein the Courts may
judicially intervene and pursue the matter and direct the parties to
undergo a DNA test. A article on legalities involved in the state of
Arizona on the referenced matter may be found at
http://www.supreme.state.az.us/dr/Text/Paternity.htm
 An informative article on what DNA is, how it works and issues
incidental to its legal validity may be found at
http://www.proadn.com/en/communique_communique_23.
"DNA polymorphism analysis means that biological kinship can be
retraced, since each individual inherits an allele from each of his or
her parents for all the loci dispersed throughout the genome. In
paternity suits, the maternal genetic traits present in the child can
be determined using DNA analysis and the presence of the remaining
traits in the presumed father can be verified. In the field of family
law, DNA analysis has improved the quality and accuracy of the
conclusions that were previously obtained through blood typing (ABO)
and through analysis of human leukocyte antigens (HLA) . It is now
possible to identify or exclude men as fathers with accuracy that the
conclusion is deemed to be certain"
Hope this helps!

Rgds,

Thackers.

thankyou for your coments