Based on my attempt to convert an non rail 3913TSW for use on my CS9, I'd say a lot of work is necessary. Also, I seem to remember either Big Dog or TercGen saying it was easier to convert the regular 3913 grips than trying with the 3913TSW.
You might be able to find that thread if you search long enough.
You might be able to find that thread if you search long enough.

I got one a couple of years ago for about the same price, also black finish, with a little wear, and two magazines.
The only thing I don't like about it is the bulkiness of the original Hogue grips. I want a set of the modified Delrin grips for it, but I'm hesitant to modify a set. I'm pretty good with metal, but not so much so with wood - and working in plastic seems a lot more like woodworking than metalworking to me.
From the picture above it looks like the CS45 and 4513TSW magazines are compatible. I've always wondered about how much the 4513TSW grips would have to be modified to fit the CS45.
Anybody know?