That will work with a war, but what about when they just demand tribute? If the first I find out about it is when Nebby (or someone) says they have bullied Jerusalem and they had to because of....., the first I realise is the message and when I look at the CS my influence has already waned.
It seems that even a DOW wouldn't restore that influence.
Correct, it will not. When you pledge to protect, all you've done is increase your CS influence resting point by 10 points. Nothing more. If your influence is above the resting point, your influence degrades every turn in the ordinary way - until you reach the influence resting point, when normal turn-by-turn influence degradation ceases.
Pledge is not a defense pact or anything similar to a defense pact. You are not required to denounce or DOW a bully or send troops to defend the CS, or gift them troops, or anything like that. But, if you fail to honor your pledge, your pledge is cancelled -- influence resting point returns to its old level, you immediately lose influence (beyond the normal turn-by-turn influence degradation) and you cannot pledge again for 10 turns.
So, what are you required to do to honor your pledge?
Everything revolves around what you do when the AI approaches you to admit that they have attacked or bullied one of your CS protectees. You will have two possible responses: (1) "You will pay for this" or (2) "this won't affect our relationship" (or words to that effect).
If you select response (2), you have shown your true colors, your pledge is cancelled, influence is lost, and can't pledge again for 10 turns.
If you select the former response, your influence resting point with the CS will remain unchanged (normal turn-by-turn influence degradation continues, but resting point remains unchanged), but you will get a negative diplo modifier with the aggressor civ (stated something like "you failed to look the other way when they abused CSs under your protection"). It is not the worst modifier, but it can add up with a repeat offender.
That's all there is to it. If you want to denounce or DOW a bully, go for it, but your CS protectee doesn't expect that, and you get no more influence by doing so.