"Disease" is a condition diagnosed by a medical profession with specific biomedical indicators.
"Sickness" is can include the state of being sick/ill physically or mentally in a person or can be used to describe a disfunctional society.
Seems to me that "I would have had a sickness" should be accepted. I reported 12/8/2017.
I believe I would have been ill would be translated as Eu teria ficado doente, which has a slightly different meaning, but I would like to see others' opinions on this.