Couldn't that be confirmation bias - if you don't think you are going to win, and it will cost lots of money - why fight?
Which is why when fathers fight for custody they may win more - because they have a stronger case.
Confirmation bias is defined as "the tendency to interpret new evidence as confirmation of one's existing beliefs or theories."
What you're describing is not confirmation bias, because there is no "new evidence" which one could interpret as confirmation of their existing beliefs or theories. If the father doesn't even try to obtain custody then they haven't discovered anything new that would reinforce their existing beliefs.