You didn't say if you are using a Unix or Windows client, but I am assuming Unix. I believe the Windows client has a way to change this, but I'm not certain and it's been a long time since I needed to use putty.
As for remapping, no, there is no trivial way to do this. VMS has a way to define what is used for backspace, but not for CR, that I can find online or in my memory.
What happens if you press control-M? Does it send a ^J or ^M? What happens if you press ^J then? Is there any chance you can run a Xwindows command-line shell (Xterm or the like) and display it locally? That would use X11 key mappings, which should be sane.