Since Royal Crusaders are considered to be troopers, they can easily be taken down by strike mechs or large amounts of other troopers. However, your strike mechs need to keep their distance, as Royal Crusaders excel at killing mechs. Since Elite Royal Crusaders are shielded, grenade troopers (which are good at taking out shields) can also be a help.
My friend and I have a Battlegrounds website completely devoted to strategies, and one of them is a Royal Crusader strategy. You can see that strategy and the rest of the site at:
Echo Base (
http://echobaseusa.tripod.com)
Hope this helped.