We camp every year, rain or shine since way before the kids were born. They're teens now.
Don't get soaked through at all. Puddle suits, especially for crawlers, are a godsend. Wellies a must. Waterproof trousers for the adults.
If you've got brushed cotton or even better, teddy bear fleece duvet covers that will help.
Sounds like you're on it insulating the floor.
Airbeds are cold to lie on, put fleece blankets below.
Youngest was 6 months the first time she went. She had a cotton onesie with a padded snowsuit on top. She got to sleep in the travel cot, more to keep her safe from her nearly nearly 2 year old sister! Blankets for if it got cold. That was July.
The older one was nearly a year when she first went, she had a fleece onesie and baby sleeping bag with arms. Again blankets on top, but if she cried she came in with me.
I was always worried about them slipping down inside sleeping bags so didn't buy them until they were about 3 & 4. Even then I didn't zip them up, just used them as duvets, until they were older. At that stage they used to snuggle up together.
The second sun hits a tent you know about it. It heats up very fast, so that's something to watch in the daytime. (Hope you get some sun!)