In chutz l’aretz, if someone other than the avel is davening for the amud (for example, on Rosh Chodesh) does the chazan say Alokeinu V’alkei Avosaino etc. (birchas kohanim) during chazeras hashatz?
In places that the minhag is not to say Birkas Kohanim in the home of the avel it also isn’t said on Rosh Chodesh, since on Rosh Chodesh even though there isn’t tachanun there is regular aveilus,. However on Shabbos or Purim, times when the avel doesn’t do aveilus in public they would say Birkas Kohanim.
Eliyahu Rabba 559-13, Dagul Mrivava 127, Yosef Daas 376, Pnei Baruch 10-26.