I had it once as a child and once as an adult. As a child, I wasn't even aware that I had more than a common cold, but the doctor listening to my lungs and one X-ray later, and it was confirmed. I took antibiotics. I don't think you need antibiotics all the time as mild cases can clear up on their own.
As an adult, I had it and it knocked me out for a good 2 weeks. The cough (both times) had phlegm but not large amounts. (I had far more phlegm with bronchitis.)
As for your story, I think it would work, but I imagine if she's in a corset all the time, it would make it very hard to have a productive cough and the walking pneumonia might get worse? I don't know.
As an adult, I remember coughing a lot. So much that I couldn't sleep. I eventually had to get a codeine to help with that. It's kind of a catch 22. When you get run down, you're far more vulnerable for pneumonia in general. And you need rest. Lots of rest. Yet the cough often makes it so you can't get rest. My back hurt (lungs). I actually pulled a muscle from coughing. And my stomach muscles were mucho sore.
I never coughed up blood, though.
I guess one thing I would look at is how doctors of the time might have been able to differentiate between tuberculosis and walking pneumonia? If this character had access to a doctor . . .
Both times I had walking pneumonia, the doctors seemed to know just by listening through a stethoscope that I probably had pneumonia. The X-rays just served as confirmation.