For some reason, I can't use the double-click search with JEI while in a LT multiblock storage, and InventoryTweaks' shortcuts don't work. For instance, I can't space-click or shift-click-drag to move more than one stack at once, either in or out.

LT uses the gui-api of CreativeCore which has it's own inventory system. InventoryTweaks does not support it.