In Nitei Gavriel on Hilchos Aveilus (cheilek alef) perek 117 he notes that when davening Mincha by the house of an aveil, there are mixed minhagim.
The case for saying we do, appears to have more support (refer to footnote 3):
Pri Megadim, OC 131, Eishel Avraham, S"K 10 notes because "אין מידת הדין מתוחה" - "we do not invoke the ...
As far as the Mitzvah of Pru Uruvu, the Minchas Chinuch in Mitzvah Alef has a long piece on the topic
Here are 2 excerpts that show that he holds although the children are not considered "sons" they are considered his "offspring". With this differentiation, perhaps you could apply it to ...
This is indeed the custom as recorded by the Rema in SA YD 376:4
The mourners recite Kaddish even on the Sabbath and Festivals, but
it is not customary for them to lead the services on the Sabbath and
Festivals, although there exists no prohibition against [this]
Commentators note the conflict between the elevated level of joy in the prayers and ...