I just figured out how to keep the battery module plastic frame from flexing inward, to keep the four connection pads on the frame in solid contact with the kindle battery jack:

Rotate the LiPo 90-degrees!
A stroke a genius, eh?
EDIT: Those four pin labels that are "almost" readable, are (top to bottom in above photo) "- C D +", where 'C' and 'D' are almost certainly "i2c Clock" and "i2c Data".