Pitch Shifting can only get you so far.
There is a Difference due to the Bagginess of the Strings when you Tune Down that a Pitch Shifter is not going to Match.
I have a Gibson Dark Fire Les Paul, which is their "Robot II" Technology.
This works quite well for Dropping to Drop C, Back to Eb, etc.
The Guitar took quite a bit of setup and a good study of the Manual, because I wanted to shut-off ALL on board EQ and Default all my Coil Switching.
Additionally, I Learned all my "Tunings" to the 1st Bank for easier access live.
This works quite well, because I can get from Eb to Drop C in 5 Seconds on average and the String Tension is how it should be.
These Guitars are Referred to as "Gibson Robot Guitars", there are several different models including the Les Pauls.
I had a Les Paul "Robot I" which I later sold on eBay, because while it worked well, it took a good 30-45 Seconds for me to change tunings.
A factory Preset that would be close to what you would need if still trying to do the Pitch Shift is around 75 in Firmware 9, it is called Down Full Step.
That patch is "Voice 1 Shift" at -2. You would want to take it to -5.