I haven’t had a chance to see the חפץ השם yet, but I do plan to check it up.
I think your קושיא is really interesting. How are we to reconcile the גמרא of אפילו חרב חדה וכו with the מנשה of תפלת שוא?
In addition to the מעשה רב from Rav Shlomo Zalman ZT”L, I would like to point out that רבי עקיבא איגר in הגהותיו לשלחן ערוך או”ח סימן ר”ל says that in a case where there is no medical chance of survival, one should not Daven for it, and that it is forbidden to ask Hashem to perform a miracle (his specific case is a בן ח’ חדשים which the Gemara says can not survive and it is אסור לחלל שבת עליו. Thanks to the miracle of modern medicine, such cases routinely survive and thrive nowadays).
I’ve only had time to do precursory research as of right now, and I haven’t found any sources on this yet. A thought which occurred to me is that perhaps the מחלה of חזקיהו wasn’t [yet] at a terminal stage. The image of חרב חדה is perhaps different from illness because of the element of בחירה.