Security at hospitals that I have been in isn't that great. While in training, I had 2 6 week clinical rotations at Greater Southeast Community Hospital in DC, first was Internal medicine, then I was somewhere else for a 6 week rotation, then back at GSE for 6 weeks of L&D. During the time between my rotations, someone came into the lobby and started shooting, and I believe someone died (please keep in mind that was 23 years ago, and I have only 1 thing shorter than my memory, that I will not go into out of respect for DC's policies). Now, not only was this in the worst city for handguns (1987 was long before Heller), the hospital was in one of the worst sections in DC, if you can imagine THAT! I drove past "Street Corner Pharmaceutical reps" on my way in and home. Rough area, but I dot a really good medical educational experience there, especially in internal medicine.
As far as your situation, if you feel that you absolutely must have a weapon, ask if you can wear a fanny pack with the scrubs. They may say no, but if not, you can get one set up with a holster (they even have more "touristy" looking ones, that just don't look like you're carrying. There's also Smartcarry, which you can try, and see if you get printing. But scrubs are made to be loose fitting, unless you get the ones that are more form fitting. Personally, handguns aside, I always go for comfort. Obviously, you can go with the 22 in an ankle holster as you mentioned, but it should be a last choice, but certainly is better than nothing (remember the first rule of gunfights- "Have a gun!"). Perhaps you can scrounge up enough cash to get a small used 9mm, and a new ankle holster or smart carry.
What ever you decide, Congratulations on the new little one. As someone who loves children, and always wanted to be a father, but I had a first wife who didn't want one, and a wonderful second wife who can't, so I'm a tad jealous, but very happy for you. Best of luck!