If you're after specifically barrows pieces, the all you really need to do is kill the brothers. You can skip the mobs inside. You can even not kill the barrows brothers you don't want specific pieces for to speed up the time until pieces you actually want (though it'll be slower than killing all brothers for any barrows item as you're increasing the drop rate for a piece by excluding brothers). If you really want to speed it up, you can use strange lock picks to force any door open.

If you did want the most efficient path for chests and still wanted to min max the reward counter. I'd suggest killing 3 skeletons until champion scroll, then killing 2 skeletons and 1 other thing (rat or worm). This, alongside the diary perk, will yield max runes without awarding bolt racks.