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I'm aware of the options. 4 is not really a thing, though, is it? You cannot prevent someone from doing work on their own property. All the adjacent owner can do is dissent, and employ their own surveyor.
Either way " I tried to use her surveyor as she instructed them before we had even instructed our own but they refused." - Amir-Siddique v Kowaliw suggets that they could be liable for some of your ow surveyor's fees (IANAL though, so pinch of salt).
Not sure if you have had party wall stuff done before, but there are basically 4 options.
FWIW I tried to use her surveyor as she instructed them before we had even instructed our own but they refused. All a bit fishy IMO.