As you write, Yom Hashoah was controversial from its inception. The date that was chosen during the month of Nissan is a day that is normally forbidden to have a day of mourning on. That is why many people keep the mourning of Sefiras haOmer from Rosh Chodesh Iyar onwards. Why do some people keep the Sefirah from the beginning? Because we are commemorating a calamity that occurred during that specific time which makes it permissible.
The date for Yom Hashoah was chosen as it was the date of the uprising of the Warsaw Ghetto and the secular government felt that it was a very apt date to commemorate the Holocaust. Whilst it is true that the twenty seventh of Nissan is also a specific date - it is a specific date for one act that took place within many years of pain and anguish for the Jewish Nation during the Holocaust - the same way ths date was chosend a different date that did not conflict with Halacha and Tradition could have been chosen easily.
The Orthodox community has always used the tenth of Teves as the date to commemorate national disasters other than the Destruction of the Batei Mikdah. In fact, the Chief Rabbinate of Israel in the first years of the State's existence, designated the Tenth of Teves as the Yom haKaddish haKlali, as the day for each person who is not sure of the actual date of the deaths of their dear ones to recite Kaddish for them.
