What is a "tittering order"?
Given the context of a regime on the verge of collapse, it looks very much like a typo.
I strongly suspect the intended wording was "their loyalty to the teetering order."
From Merriam-Webster's definition of teeter:
1 a : to move unsteadily : WOBBLE
1 b : WAVER, VACILLATE
// teetered on the brink of bankruptcy
2 : SEESAW