In the version of ashes i play Sympathy Pain reads "You can play this card >after< your Phoenixborn has received damage.[...]"
The FAQ describes when effects get resolved that trigger when something takes damage.
Step 1: A unit or Phoenixborn receives damage
Step 1b: Effects that trigger >on< receiving damage
Step 2: Place wound tokens on the unit or
Phoenixborn equal to the damage that they have
Step 2b: Effects that trigger on a unit’s destruction or
a Phoenixborn's destruction happen now.
If you are strict you can not use the Rule 1b because it does not have the exact same wording and talks about simultaneous effects.
Sympathy Pain is explicitly worded to be playable after you already received the damage so you have to survive in order to do so.
Obviously you may have another opinion but they promised an updated version of the timing rules for the next expansion and we will probably be able to see what the intention was.