So - just addressing the DNA part of your question - if I am reading what you wrote correctly, you would be trying to use DNA testing to show that your great-great-great-grandfather is the father of the woman's great-great-great-grandfather.
If you use an autosomal DNA test (like Ancestry DNA), your match is distant (4th cousins - 1X removed - your gr-gr-gr-gf is her gr-gr-gr-gr-gf), so you would (at best) be dealing with one or two DNA segments that you share. Just having a matching segment or two with her is not going to be enough to prove that the DNA came from this specific ancestor.
Also - the odds of two 4th cousins matching is less than 50% and the odds of 5th cousins matching is just around 12%, so the lack of a match between you and her does not rule out a possible relationship.
Honestly - in terms of DNA, your best bet would be to try and find a male descendant of your gr-gr-gr-gf and a male relative of the woman's family and do Y-DNA tests (FTDNA has a 37 marker Y-DNA test for $169). If the male on your line matched at 37 markers with the male on her line, this would be solid evidence of the family relationship, since a 37 marker Y-DNA match (37/37, 36/37, 35/37 even 34/37 would catch my eye) between two men is strong evidence of a shared paternal ancestor within the past several generations.