How can I use a BlueSMIRF Silver (RN-42) as a HID Keyboard (on IOS)?
I should be able to set and use the HID profile (from the default SPP)


Unfortunately, your assumption is incorrect. From the same documentation you quoted, the previous sentence says:


  To use Roving Networks’ Bluetooth HID profile, you must use a special
  build of firmware, version 6.03 or later.


Poorly phrased, this does mean that you need to have two things:


A special HID firmware 
Version 6.03 or later


Your BlueSmirf Silver only complies with the second condition; it does not have the special HID firmware (also note that the Rovnix command reference does not address a specific variant of the BlueSmirf, but speaks generally about all of the variants).

The SPP protocol (Serial Port Profile) is used for emulating a serial cable, and functions as a substitute for the physical RS-232 connection, but cannot stack pure HID profile on top of it. In order for you do talk HID, you will need the HID firmware. 

As a side note, the HID version comes with SPP and you can switch between the two modes interchangeably, I have been using the HID version for a while now and it works fine (including the pairing with iOS). 

Another misfortune is that it does not appear that Roving Networks officially support any re-flashing of the BlueSmirf firmware, so unless you are willing to reverse the hardware, this is off the table as well.