Is it possible to fail Cold Call Killer?
Solution 1:
Alright, finally tried it: if you don't get enough progress on the trace, you will trigger a game over. You'll have to restart from Zsasz sending you running for the payphone, in much the same way as if you don't pick up the second payphone before it stops ringing.
Solution 2:
I think I may have failed it. :/ In my stupidity I didn't realize that you actually have to MOVE the cursor around to trace the calls - I only figured this out on maybe the third call. It really looks like he's not interested in playing his game any more.