I'd use nylon strapping for the straps, that velcro shut at the back. I wouldn't use craft foam unless you reinforce it, as it'll go limp. Something stiffer (whether you use plastic or somesuch), will also help to hold it all in place, you'll have to adjust it less.
And as weird as it sounds, if you want to keep the straps in place, try getting carpentry tape. It's a double-sided tape that they use to bond the edges of carpet and it's similar to hollywood tape or toupee tape, but strong as hell, and will hold up for a whole day.