Febreze and things like that just mask the smell, and you're right that they can cause other problems.
Use dish soap just to get rid of any food residue or visible dirt. If there needs to be any scrubbing involved, you can also use baking soda, then rinse clean. White vinegar is also terrific, and if you rinse it, it won't smell afterwards. If there's a smell inside the pockets, dip the whole thing in a bucket of vinegar & water (about 1 cup vinegar to a gallon of water, or more if you think you need a little extra power). Then stuff some newspaper in the pockets to let some air in there, so things dry without creating more smell.
Even if the "finished product" smelled a little vinegar-y, the vinegar is better for your friend than the musty and mildew-y smell!) And yes, let it sit in the sun when you are done, but only when it's good and clean.
I only use baking soda or vinegar for cleaning pretty much anything - plus a basic dish soap for something oily. You don't need any of those other high priced cleaners which are just chemicals or a version of what you would make yourself with the basics.