Ok, so I have had a bad weekend and am having a bad day. I was using humour in an attempt to cheer up both OP and myself, but obviously, it didn't land very well.
The chihuahua did bite me. She was physically shaking from the effort. It didn't cause any damage more than a minor bruise and a bit of a sore thumb. Yes, they have sharp teeth but they don't have the jaw strength to seriously puncture living flesh with them.
I would imagine they can tear something as thin as an eyelid. A scraggy fingernail could tear an eyelid if you were that unlucky but you'd have to be either seriously unlucky, stupid enough to stick your face in an unhappy Chi's face or a combination of both for that to be a risk.
I do agree, however, that this is a serious issue. DD uses "gremlin mode" as an affectionate, humourous description of unwanted behaviours that we do address immediately. I would never allow her to keep a dog that was "fulltime gremlin mode" without addressing its issues. Her current dog only goes into "gremlin mode" if she thinks someone is going to take a shared toy she has randomly decided belongs to her and her alone or if the bigger dog accidentally steps on her.
I also agree that if a behaviourist says the dog cannot be helped the dog should be pts on welfare grounds. Dogs have an instinct to survive. They do not start or escalate conflict unless they feel they have no other choice/their life depends on it because they instinctively know injury could mean death. An injured dog couldn't hunt, for example. So a dog who is constantly in a state of high arousal is a dog who is constantly in fear for its life. That's not a happy dog. It would not be fair to allow it to continue.
But it is highly, highly unlikely that behavioural therapy couldn't address the matter. What I meant by if OP feels she really cannot manage, is that if OP feels she doesn't have the time or means to offer behavioural support she should look at rehoming to someone who can offer the dog the right support.
Fear aggression is easier to address in smaller dogs, especially dogs as small as chihuahuas, not because they cannot cause serious harm, a qualified behaviourist would never push a dog to the point it felt it needed to bite, but because their exercise and stimulation needs can be met inside the house/garden if needed. Outside walks and off-leash time is great for welfare and chihuahuas should be walked daily most of the time, but in this case, it is causing more harm than good.
Meeting the needs of a larger dog without leaving the house and potentially encountering a trigger is much more difficult.