A sea anchor is big and essentially stops the boat. For a 46+ catamaran it seems like it would put a lot more stress on the boat than anchoring because the anchor chain's weight acts like a shock absorber in the waves. If a wave jerks the boat really hard, the anchor chain will straighten out and absorb a lot of the shock before it is fully tight.
This isn't true with a sea anchor on a rope. The sudden jerk of a breaking wave will be transmitted to the boat almost immediately because the rope or chain to the sea anchor is already tight. The wind and waves in a storm will put a lot more pressure on the boat than normally encountered at anchor.
Where is the strongest place to attach a sea anchor? cleats? windlass? winch? No matter where you put it, you should be able to let out a few inches of rope every so often so it will chafe in a new place. With either a drogue or a sea anchor, you can probably tie off the helm and go inside, or use the autopilot if it's available and can handle the conditions.
A drogue, because it is smaller, will have much less pressure, maybe by a factor of 10. That's why the boat moves. The purpose of the drogue is not to stop the boat. It's to prevent surfing so one of the hulls won't dig in at the trough of a wave. Also, with some forward motion a catamaran will ride on top of the waves better because the speed of the waves relative to the boat is less.