I really doubt they would refuse the visa for the infant child of two PRs. That doesn't sound like something that fits in their mandate.
But they do that all the time. There have been several couples who wrote on this forum seeking advise or sharing experience after having been denied a TRV for their baby. In a couple of cases, somebody managed to get a TRP for the baby instead but the embassies often don't tell people that is an option so after being denied a TRV, the most common solution is that the father goes to Canada alone to sponsor the child while the mother waits with the child.
What about the health coverage of the infant while sponsorship is in process?
In most provinces, a person with visitor status has no right to get health care but I have heard a couple of cases from Alberta where they covered spouses who were on visit status and in the process of being sponsored for PR. Alberta also has first day coverage if you move from another country while some other provinces make you wait for 3 months. If you have decided which province you are going to, you can contact their health care and ask.
Just read your reply now as I was about to post that you are going to Ontario. In that case, you will all have to wait for 3 months to get coverage and your youngest will probably have to wait until getting PR but ask OHIP about it.
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