[KensRedZed]

Since you're not trying to break land speed records. I might use a small piece of hard wooden block with wood screws. Cut it down to size.

Do you have a length of scrap plastic, or Evergreen square styrene tubing to glue across the body?

I'm not a big fan of popsicle sticks. That's were I start to wince.

I would start rummaging through your materials to see what you've got. Sometimes a chunk of a chassis from another static model might work. Or a nice, small piece of hardwood?

Wood epoxies real nice to the inside of the body...

Best of luck,
Ken