The phrase dysfunctional quiver as used here means an unhealthy, problematic, or defective sign of weakness, instability, or disorder (shakiness, lack of solidity) within the royal family--a precariousness.
It is not clear if the author meant to describe only the royal family in this way, or the royal family and prewar Britain as a whole.
Yes, quiver means a shaking, shakiness or tremor, it can indicate weakness or infirmity, instability, disorder, etc.
Here a dictionary shows one sense of shaky as precarious:
The definition of tremor here shows how several of these meanings can be synonymous:
Definition of tremor
1 a : a trembling or shaking usually from physical weakness, emotional stress, or disease
b : nervous excitement
2 : a quivering or vibratory motion; especially : a discrete small movement following or preceding a major seismic event
3 a : a feeling of uncertainty or insecurity a tremor of hesitation
b : a cause of such a feeling
Dysfunctional means, essentially, something wrong; something not behaving or working normally