What screen capture tools can copy the image path to the clipboard after a snap?
You could just assign a shortcut to a script like this:

#!/bin/bash

f=~/"Desktop/$(date '+%Y%m%d%H%M%S').png"
screencapture -i "$f"
echo -n "$f" | LC_CTYPE=UTF-8 pbcopy


See man strftime for a reference of the date format.