Pain is complex experience that doesn't necessarily involve a physical stimulus. It's kind of like asking if we can measure objective happiness.

We do have tests like the ice water paradigm you mentioned but they are limited as people may withdraw their hands at different pain thresholds (eg if one subject is apathetic about the study they might withdraw their hand sooner than another who is invested and is willing to withstand more pain). There have been more recent attempts at measuring pain perception in the brain. Currently there is research into identifying which regions of the brain activate in response to pain and seeing if we can predict the amount of pain someone is experiencing based on their brain activity in those regions. But generally this requires EEG or MRI which can be expensive and time consuming.

Advances in pain assessment: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6979466/

Phantom pain: www.mayoclinic.org/diseases-conditions/phantom-pain/symptoms-causes/syc-20376272