Now that's a big FAT boom.
If you make the space underneath the fat / tallow a food stockpile, set to accept both, there will be no need for immediate moving. Then you can designate stockpiles for fat / tallow in desired locations, and set them to take from the combined stockpile. You will have to do this with one of the two first (preferably the fat, since I think that rots even when in a food stockpile), because it is not possible to have 2 stockpiles take from the same stockpile.
I'm not sure about tallow rotting. I always have it stored in barrels, and use it up to cook / make soap rather quickly most of the time. Never seen it rot, except when left in the butcher's shop for too long before being hauled.
I've never seen prepared foods rot, except when left in the kitchen too long. I do not store prepared foods in barrels either, and dwarves do path over it.
Maybe it has to do with my cats, who are quite effective at killing vermin that spawns in my fort.
For the fat / tallow items scattered in your dining room.. If your dwarves are not hauling it, consider using the (o)ptions menu to turn off all hauling except refuse hauling, and order the items in your dining room to be dumped. Dumping items seems to have a rather high priority. In any case, higher than fat / tallow hauling. This, ofcourse, does need an active garbage dump to a place where miasma is no problem / atom smasher.
On a quite unrelated matter: I can see, on your screenshot, that quite a few of your rooms have only a 1x1 entrance (single door). If your FPS starts dropping too dramatically, widening those could give significant improvement.