I practiced in a hospital and outpatient setting for 10 years as an Occupational/Hand Therapist prior to my current profession and I wrote with a fountain pen during those years. I was never questioned for using a fountain pen. People (including physicians and the director of nursing) thought it was a neat thing and a topic of conversation. They don't have the knowledge that we do about ink properties, so they assumed it was just like a regular ballpoint or rollerball as ink goes. I never encountered the "ink police".
This was in an era where documentation was going to computer so there were fewer paper forms as time went on. However, there are a few things to be sure of with paper med records and some things that really don't matter. Here is what matters:
1) Black - all hospital documentation must be in black
2) Must be legible - so good handwriting and no feathering are important
3) Must be able to be copied (well, black ink copies just fine)
4) Must be permanent. You won't be writing prescriptions, but you still are creating a legal and medical document.
So this kicks out Iron Gall inks because they aren't black - and a blue black that turns dark black like Diamine Registrar's won't cut it in the short term. You will be questioned and accused of writing in blue. They won't care that it turns black over time. I don't think you need to worry about Iron Gall ink eating through the paper. The patient (and you) will be long dead before the ink would do that. Even so, today's formulations of IG most likely will not cause that damage.
I would also avoid inks that shade to avoid a questioning eye. You want a clear, crisp black line.
In my mind, you want one of these two:
1) Sailor Kiwa Guro - a carbon ink that absolutely will not feather or bleed through. It has a dark line and it is permanent. It will not smear when dried.
2) Platinum Carbon Black - similar thing as Kiwa Guro from what I hear.
Unfortunately, Montblanc Permanent Black smears even when dry on coated papers so it is out of the running. It also can bleed through.
Noodlers Black is out too, because it can bleed through on some papers.
If I was still in a medical profession, I would use Sailor Kiwa Guro with a Lamy 2000 fine nib. Why this particular pen? Because it doesn't look like a fountain pen to most eyes.
Even so, you will still want to carry a rollerball along for those times when you have to write on those carbon/pressure forms that occasionally pop up in a medical record. The Kiwa Guro will not show through anymore than a ballpoint or rollerball. To be honest, a rollerball can bleed through.
I think someone's statement above is good - take a writing sample to your boss in the pen of your choice with Kiwa Guro, written front and back. Get it approved and enjoy!