You can use the Doodle apps without needing to change to an adult profile, but you will need to share the content with your child's account.
To do this, please follow these steps:
In the adult profile, head to Amazon Apps and find the Doodle app you want, download and install it.
Open the Fire for Kids app, and open the profile of the child you want to grant access to.
Tap "Add Content" and then pick the category, e.g. Games & Apps, and select the apps you want to allow the child to see.
Hit done at the bottom of the page.
Sign back into the child's profile to access the app.