DD1 didn't like being cleaned. I turned it into a game. We now 'freshen up the buttercup'
So (please don't laugh at me, because it works, but it is cringeworthy just typing it)
I sing a song:
'Freshen up your buttercup'
'Freshen up your buttercup'
'Freshen up'
'Freshen up'
'Freshen up that buttercup'
All very relaxed, but I look like a hawk to see if she is 'dirty' inside. If I have seen that she is cleanish, then I just do one rendition of the song.
If I see that she is quite 'dirty' I repeat the song, and that gives me another chance to wipe.
The other thing you could try is to go to the Chemist and buy a tub of 'Aqueous Cream'. It is very cheap. It is a semi-liquid, soft gloopy soap substitute.
The beauty of Aqueous cream is that you don't actually need to wipe it off carefully, or rinse. You can use it in the bath, like soap, or you can rub it in to sore dry skin.
So, although you want to eventually encourage Lucy away from touching her private areas, because they will get sore, and she is smearing, you could do the opposite to desensitise her.
Ask her to give it a quick wipe. That will get most of the poo away. Then let her smear lots of cream on her bits (IMPORTANT: put some AC in a separate bowl so that she is not dipping pooey hands into the clean tub of cream). She will think she is playing, but she is actually working the cream into her delicate areas.
Then, ask her to wipe it off, and tell her she can have another go. Hopefully, the pull of getting another go will encourage her to wipe it off.
The second time, she will be quite clean, and the AC will just be moisturising and protecting her skin.
Do you use Sudocream? That is a good barrier to put on after cleaning, so that next time she soils the poo isn't contacting her skin.
You need to break the assosciation between pooing and being sore afterwards. It might be part of why she doesn't tell you she needs to go.