I think your situation is a little complicated. As a US citizen he does not need a visitor visa he can fly in on his passport. As a PR you or your wife can sponsor him to be a PR but according to current processing timelines it would take at least 14 months for him to get it and a US citizen can only live in Canada for 6 months. Also I’m guessing that your child was born after you filed and received PR?
It’s possible that Canada might give you special permission to keep the child in Canada while your application is getting processed but this is a pretty special case. My advice is this is to reach out to a immigration lawyer at the earliest since the “Family Class” applications can only be filed from outside Canada.
Yes he would not receive state sponsored health insurance, but I think the bigger issue here is permission to stay in Canada. Applying for the child is a straight forward process what I am not clear about is on what status would your child live in Canada while the application is being processed.
In our case we waited and delayed our application (informed them via email) since we were expecting a child and only filed after we included him on the application.
Details on sponsoring your child
The sponsorship application process