You don't wanna do that.
When the CIC sent my wife the 2nd medical exam request (via email), they sent a PDF file (IMM 1017) already pre-filled with information such as her date of birth, UCI, file number, home address, email address, and phone number. And she was instructed to bring it to the panel physician.
And most importantly, the form also has a unique ID called "IME" (consists of 8 digits). Underneath the IME number is a barcode.
I don't even know what it stands for, but this IME number wasn't mentioned in the Singapore AOR letter, and the first time my wife saw it was when she got that 2nd medical request.
So I doubt it's a good idea to do the 2nd exam without having that special form.
It's best that your wife waits till the CIC requests her to do the 2nd exam
.