I've witten this before gerald_clark has posted on which I have to agree. Anyways...
Have you tryed installin Core manually? This seems easy, too, since mostly everything is set up already.
You could boot into Puppy, mount the TC .iso image and the partition you want to install to, copy respective files ...
+/tce/
|- onboot.lst
|+/boot/
| |- core.gz
| |- vmlinuz
|+/optional/
| |- "all *.tcz goes here"
|+/ondemand/
| |- "will be used later by Core"
... and edit your boot loaders config (the same which is used to boot puppy) accordingly.
Something like this for syslinux.cfg
LABEL core53
MENU LABEL Core 5.3
KERNEL /tce/boot/vmlinuz
APPEND root=/dev/sdxY initrd=/tce/boot/core.gz
NOTE: change root=/dev/sdxY to your needs.
You can also install Core to the same partition where Puppy is installed, if there is enough free space.