Personally I would do it when she's older (mostly because we got rid of DD1s dummy when she was a little over 2 and it was a doddle)
I think there's an awkward age where babies are old enough to remember the dummy and know they want it, but not old enough to be reasoned with.
We only allowed it at nap time and bedtime. We did the dummy fairy routine.
If you don't want to let her CIO. You could try the baby whisperer's pick up/ put down and shush pat methods. Basically you rock LO until drowsy, and then place in cot and pat until asleep. If LO starts crying you pick up and start again. You pick up as often as needed, and put them down again as soon as they are calm.
If you haven't done that kind of thing again, and as you'd be taking away the dummy I imagine it would be quite an exercise in patience and persistence, but if you stick with it, then it should work.
ETA: for the rash DD has had a rash around her mouth that I'm pretty sure was sparked by the dummy a couple of months before we got rid of it. Dr recommended vaseline jelly around her mouth as a barrier before eating, and bepanthen ointment at bedtime.