Something just occurred to me that hadn't occurred to me before. Renee is easily misdirected. We see Bella do it to her in Eclipse, and even Bella is a little dismayed by how easy it is to do. I'm sure the Cullens, especially Carlisle, who is so reassuring and confident, can misdirect Renee for a long, long, time during Bella's illness and recovery. They are all accomplished liars, both due to their lack of tells and their long history of subterfuge amongst humans.
Charlie, on the other hand, is a police officer, and very single-minded about Bella. He doesn't automatically trust other people's opinions over his own like Renee does. He keeps his own council and forms his own ideas more often than anyone else in the story. So where Renee will take Carlisle at his word and assume that Bella is in good hands, Charlie has to see it himself to believe it. This explains why Renee is unconcerned about Bella's mysterious illness, and Charlie gets suspicious.
However, something will begin to bother Renee eventually, and she'll figure out that something is up. Her intuition will eventually come to bear. I hope they've come up with a good story long before that happens!
Thanks for listening - Ariadnee