Oh geez, I'm sorry.... I totally mis-read your project - I was thinking "mattress" for a doll bassinette. The 3/8" craft foam would not be at all suitable for the walls of the bassinette, only the mattress part.
I did find out that Flex Foam is a little thicker than Bosal In-R-Form, but they're about the same for stability, so I would go with the Bosal - it's available as a fusible, but it does cause some buckling/wrinkling on the fabric surface. You can stabilize your fabric with woven fusible and then apply the fusible foam and the wrinkling isn't as noticeable.
On some domestic machines, I would cut the foam to fit inside the seam allowances and baste it to the individual pieces either with big stitches, fusible web or spray adhesive to the individual pieces.
Some people leave foam in the seam allowances, but they zigzag or serge around the edges to compress the foam. If your machine will handle the heavier seams, the project will be sturdier.
Sorry I swung and missed on my first answer. 🥴