Do not confuse Bump Boxes (the more common name = Bounce Box) with a re-supply box.
Bounce boxes are for those items that you will or may need at some indeterminate point on the trail AND which you CAN NOT reasonably expect to be able to purchase locally. INCLUDE such things as prescription refills, spare eye glasses or prescription sunglasses, memory cards for your camera, unique spare batteries, a spare platypus bladder or a platy repair kit, orthotics or special footbeds, a pair of new socks, blank mailing labels, tape, etc.
DO NOT INCLUDE such things as food, snacks, fuel, TP, soap, toothpaste, baby wipes, chewing gum, etc. Those items, and guidebook pages for the up-coming sections (until your next re-supply box) go into your re-supply boxes, or to the extent possible, purchase your supplies locally and support the trail town economies.
I suggest you use the post office's Fixed Cost Priority Mail boxes - there are 2 sizes - for your bounce box rather than the much larger and clumsier 5 gallon pail. Save that for your re-supply package to Muir Trail Ranch (that's the only packaging they will accept).
REMEMBER - TO SHIP A BOUNCE BOX TO A PLACE WITH NO OUTGOING MAIL SERVICE IS TO BE STUCK WITH IT. Plan ahead - carefully!