In shul yesterday, we said "Av Harachamim" before Ashrei. We also did not bless the month of Tishrei. I know that there are various reasons why we aren't mevarchim the chodesh Tishrei and I figured that since, according to my siddur, we don't say Av Harachamim on a festive shabbat (when Tachanun would be skipped were it a weekday) as signified by blessing the month, the lack of said blessing would require saying the A"Har -- the two seemed intertwined. During sefirah, when, regardless of blessing the month, the entire tone is sad due to bloody massacres, we say A"Har even when we bless the month (from the commentary in the Artscroll siddur) to reflect this pervasive sadness.
But the shabbat before Rosh Hashanah is no more sad than any other. Our reasons for not blessing Tishrei are not because the month doesn't start -- the shabbat should be as festive as any other on which we do bless the month! In fact, the Koren siddur implies in the English instructions to omit A"Har ("It is also omitted on a Shabbat before Rosh Hodesh (except for the Shabbat before Rosh Hodesh Iyar and Rosh Hodesh Sivan)"). The Hebrew makes the inference more tenuous, labeling the day not "The Shabbat before" but "Shabbat Mevarchim" though this doesn't clarify whether the Shabbat before Rosh Hashanah is a "Shabbat Mevarchim on which we don't say the blessing for other reasons" or "not a Shabbat Mevarchim even though it precedes the new month."
Is there an inherent connection between the saying of A"Har and the saying of the text of birchat hachodesh? Is there some other, external reason why we DO say Av Harachamim on the Shabbat preceding Rosh Hashanah?