” One of the problems the Gedolim have with the date is the fact that it’s in Nissan when we are not allowed to mourn”
thats a pretend reason
If you think long and hard you will come up with a mourning period that takes place during Nissan commemorated by many members of Klal yisroel that lasts 33 day…
The real reason is WTP’s reason above. furthermore it is meant to lead in to yom ha’atzmaout which occurs a week later as a sort of Galus/geula contrast whcih of course the charedi world opposes.