This is the extract from FCOM (30 May 2012 for A 330) :
Note:
1. On a damp runway no performance degradation should be considered.
2. It is not recommended to take off from a runway covered with more than 4 in of dry snow or 1 in of wet snow.
However #1 is removed at new edition of FCOM. I guess Airbus did right.
Damp is not a certified runway condition therefore no info about that; however one thing is sure, "Damp is not DRY". Accepting wet calculation in Damp is at conservative side and safe.
Not current one but the place where I worked before, it was a written company procedure accepting damp as wet.
Finally, it is not a written rule but, IMHO, your Captain's discretion is right.