Yup, secure it with straps and keep an eye on it along the way. I've had my 12ft kayaks out the back of the bed. Strapped down for its life, but we get up on the highway and it makes it. You'll be just fine
Is your bed fairly new and in good shape? If not you may want to consider buying a new one wherever you are going. That's what I'd do at least.
This is kind of what I was thinking. I know mattresses can be expensive, but I don't think I'd try to haul one 1300 miles without a cargo van or U-haul truck if there were any other option. You might get a few dollars for your current one on craigslist or something, and buy a new one when you get where you're going. The cost of a new one pales in comparison to potential liability if the thing flies off on the interstate.