Line managers manage. Hr advise. People who are saying it's none of the op's business and Hr will deal with it are wrong ime.
The medical details are none of the OP's business. If the OP is the line manager (which they haven't actually stated) then they will need to know adjustments required and reasonable work expectations which may include a high level title for the illness/condition. Medical details are entirely up to the sick person to share as its private information.
eg X is being treated for a condition which requires them to attend a series of medical appointments, its possible they will be unable to attend at other times due to the effect of the treatment. You should anticipate A and B adjustments.
Anything more is not the LM's business and actually not useful unles the LM has a side line in oncology expertise.
I think it's a bit mean spirited for you to want to re-schedule her two non-working days to fall straight after her treatment days (so the company is not impacted by her absence).
I'd be astonished if OH sanctioned it. This kind of scenario is more likely to result in recommendations to be guided by the sick person during the treatment and to be prepared for both "well" and "sick" scenarios until a pattern has settled.
When I was first diagnosed with cancer I didn't have anything in writing.
I've had quite a few staff over the years affected by life threatening and debilitating illness. Its not uncommon for them to have very little in writing early on and also to be disoriented, frightened and sometimes appear bullish or in denial. HR are the people to manage this until they are ready to share more.
I've never considered it appropriate to stalk them on SM and post to random forums about them lying and skiving because "they don't look sick".
Still, I dare say this will disappear for privacy reasons soon.