According to his Wikipedia page, his downfall was the loss of the capital island of the Falkland islands. He was a true military dictator, abuses and all, but he might have understood that losing the Malvinas would be the end of his regime.

In that case, going against UK in an unwinnable war was a way to preserve his regime, as long as no bombings were made targeting him, so he could still protect his life. UK is not as ruthless in war as, say, USA or Israel, IMO that was a calculated risk to prolong the end.