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Instead of using Puppet Warp over the layer whit a vector mask, which only will transform the selection and not the path

i'll suggest Edit>Transform path>Warp "transforming the path instead" which will get you to the same result as using puppet warp on the layer, but this time you will be transforming the vector mask

Later you can fine tune the paths using the paths or direct selection tools

This probably will seem confusing but it will work.

also i included a link about transforming vector mask

Photoshop editing vector mask

Good luck