I was wondering if it is possible to initiate chdk on the N without using the lcd screen.
Yes it is.
CHDK is loaded (initiated) on every supported Canon P&S camera model in one of two ways - the
firmware update method or the
SD Card lock switch autoboot method. Details of how this works are
here.
For the Powershot N, the both method are available. The autoboot method works when the micro SD card slot cover is left open (or the little bump on the cover that engages the internal microswitch is removed

). So the answer to your question is "Yes - you can setup the N to automatically load CHDK when the camera starts. You can also activate the CHDK GUI with the middle switch on the side of the case so you don't need the touch screen for that. But without the touch screen there is not a lot more you can do with the GUI.
Also, CHDK can be configured to run a Lua (or uBASIC) script at startup - no LCD touch screen interaction is needed there either.
And if you use CHDKPTP to communicate to the camera from your PC, you don't need the LCD or touchscreen at all.
This would be very interesting as I would try to remove the lcd to lighten the N. If this worked, I think it would make the N the cheapest & lightest chdk camera that can shoot 2.3 continuous 
You might be able to remove the entire LCD & touch screen module with its enclosure. It's connected by just two little hinges and a small ribbon cable. However, the camera's DIGIC4 controller might notice the missing electronics and throw a fatal hardware error message and shut down. The only way to find out is to dissassemble it and try I'm afraid. Fortunately the camera is assembled with small phillips screws so dissassemble should not be too hard.
I have not experimented with the maximum continuous shooting speed of the camera. Frequently the published spec is not maintainable at full resolution for very long. If the shot rate is faster than the rate the camera can convert the image to a JPG and write to the SD card then internal camera memory will fill up quickly and the shot rate will slow down. If you'd like I can do some experimenting with this?
Finally, I've noticed that Powershot N's occassionally show up on the
Canon refurb website for about $100 so if you can grab one there it's a great deal.