I didn't say they were being supplied only through the BT. Logically, if they're having to import food they must have multiple sources. Both because of how much they need, and because having only one source would be putting all their eggs in one basket and they'd be SOL if it ever got cut off. They're probably gating food from all over, including from the BT. Personally I think they're also plundering Seanchan since it's both defenseless and off the radar. (Surely Semi was doing something over there besides just killing the Empress?)

I guess conversely, of course, if the Shadow's been preparing for a long enough time, they could have just been stockpiling all along and not need to bring anything in now. I sort of doubt they have been though.