The לוח דבר בעיתו explains (regarding Rosh Chodesh Iyar 5769) that the reason we don't announce בה"ב on Rosh Chodesh is that:
אין מברכים את המתענים בה"ב בחדש ניסן או בשבת ר"ח, השבת הקרובה, כמו שנאמר (קהלת ז, יד) "בְּיוֹם טוֹבָה - הֱיֵה בְטוֹב", ומברכים בשבת שלאחריה.
I.e. In a time of good fortune, enjoy the good fortune (Koheles 7:14) - so we don't want to mention the fast (which is to atone for the sins caused by the good times) during those days which are happy - i.e. the month of Nissan and the demi-festival of Rosh Chodesh.
As in: Why dampen the atmosphere by mentioning an upcoming fast.
His source seems to be the MaHaril in his chapter on סדר שני וחמישי. (Hat tip: Joel K)