Interesting predicament to normal on visas.
Having a little think about your words, may I suggest to you that there is both a Peruvian & a Pakistani Embassy in Santiago, Chile. Contact details for both are on the Internet.
"They said that "she needed a letter to demonstrate that she is single, and you can get that letter from Pakistani embassy. "...well, maybe you could get that letter from the Pakistani Embassy in Chile???? Easy-peasy !!
OR,if you actually need another visa for her, do you think that you could organise your daughter's visa through those two...forget about China.
If that turns out to be possible. maybe you can do it from here...but if that's not possible, then you & your daughter could fly down there....3 1/2 hours flight, and you can get cheap airfares also on the internet.
I don't know anything about what type of visa she has....is it like a normal Peruvian Tourist visa ?
Hope this helps...revert if you can clarify anything more.
Retired Aussie in Peru