Is it possible to have a tilable full chest detection in this scenario?

Yes - using a 15 signal strength detector utilizing the quirk of comparators that when reading a container through a solid block, they output signal strength corresponding to fullness of the chest, unless the block receives power level 15 (when they start outputting that signal strength 15).

The barrels are empty. Utilizing hoppers instead of barrels will be a bit more expensive and less lag-friendly but will allow easier access to the second row of chests.

A slightly worse w.r.t. to lag, but much more compact way with a quirk that it only activates once the 'back' hopper fills up, not the chest (might be desirable as it will only fill up if all chests are full; if someone empties all chests except of the top one, the prior method would falsely announce the slice is full): Item frames can be fit into the same block space as comparators, and without items in them they'll produce the same 'empty' signal to the comparator reading them through the block. In this case barrels are only used as extra storage space and can be omitted (a doublechest in that location wouldn't be openable.) It costs an extra row of hoppers (you can't read the filter hoppers of the overflow-proof sorter, because being overflow-proof, they are never full!) but you get a very neat isolation of the redstone-heavy "back-end" from the neat storage space with indicators close to their respective slices and no blocks obstructing access to the chests.