ok nobody knows what you are talking about, maybe you are using a third-party shipping service instead of the post office. If you can't ship this for $5 or less then whoever wins will probably be very upset, since they have flat rate mailers you can just shove the thing into at the post office and ship for $5.
first class mail is 3 - 5 days, and is for packages under 13 ounces. This looks like it would be well under 13 ounces. Unless you are mailing it in a box full of bricks, there is now way that it would be more than $3 - $4 tops. you don't need a flat rate box for this, that's a waste of money. just put it in a bubble mailer, it will be fine.