I think your manager was wrong to push you, especially given that you had previously given documentation (unless of course, cleaning up bodily fluids was specifically mentioned in the job description).

When I worked in retail, a child vomited in the store, and the manager (we had no instore cleaner) cleaned it up no questions asked. However, I now work in a school, the first few times children vomited I did clear it up, because it is generally understood that if the caretaker is off site, that is what happens! If I refused, the other staff would have been extremely annoyed with me (and believe me, you don't want to upset school staff!) In a way it was good for me, because I saw that I could do it, even though I was panicking inside. Now I have a valid medical reason (not that a phobia isn't valid, but it is poorly understood and I have no documentation for it, I do have documentation for my new reason), of course I won't clear it up! And if my boss refused to accept the reason, I would be very tempted to quit (not just over the vomit, but the lack of respect it implies).

As for getting references and a new job, I personally would 'come clean' if you were offered a job, and explain things before they ask for references. Then you can say you have a phobia, a genuine phobia with documentation from your doctor, and that you are seeing a psychiatrist. Unless the job specifically requires you to deal with vomit, I think if they then withdrew the job offer they would be in breach of DDA or something similar.