The rest of the sentence could be translated in this way:

"In the 4 weeks prior to the start of treatment, the patient has had an infectious disease requiring either the intravenous administration of antibiotics or hospitalization"

If 既往 is being used here in a strictly medical sense, then it would have to be short for either 既往症 (which grammatically would seem to be most likely here) or 既往歴. Both of these would mean "including past conditions". Translated into English this way, it would certainly sound strange in the context, as you indicate.

Outside of medicine, however, 既往 is often used simply as a synonym for "過去", and it is used much more flexibly in Japanese than "in the past" is used in English. That is, 既往 does not need to refer only to the far past, but can refer to anything that has ceased to be true before the current, present moment.

If 既往 is being used here in that way, then I think that the parenthetical expression would mean "including cases in which the patient is now fully recovered". That is, the author would be saying that it doesn't matter how well the patient is right now; if he/she required an antibiotics IV or hospitalization anytime within the prior 4 weeks, then he/she cannot take part.

Note that the sentence is written in the present tense (有する) rather than either the past tense or "有したことがある". Perhaps the present tense was used here to maintain consistency with the other exclusion criteria, but using it to refer to a 4-week span of time makes it difficult to refer to something that existed but disappeared during that time. So the use of "有する" could be the result of bureaucratic rigidity, and the use of "既往含む" could be a bureaucratically inept attempt to try to fix the problem with "有する".