It has been a long time since playing with Odin. There were a few problems with installing programs, some used 16 bit installers, which Odin couldn't handle, there was a way to change the installer to 32 bit. Newer installers probably don't have that problem.
One trick was to unpack the installer, you'd have to Google to see if that was possible. Another trick was to install on Windows and move the install over. That meant catching all the DLLs and also registry settings.
I do remember running the Win32 putty at one point, probably an old one compared to what you're installing.
Sometimes DLLs needed to go into System32,which I guess is now somewhere under @unixroot.
Most of my experimenting was before it became a @unixroot thing. Bitwise did a lot of work on Odin, but it was all concentrated on things like getting Java or Flash to run.
The weird thing is it is hard to even figure out the directory structure since Odin was moved to @unixroot. Seems the DLLs are now mixed into @unixroot/usr/lib, where there is a kernel32.dll. Perhaps missing a needed export for your Putty install.