Every mattress you've ever seen on the side of the road came from someone like you. Hitting a mattress is like hitting a deer, it will likely total the car, and can ignite chain reactions that can kill.
This is serious business, and it needs to be immediately clear to every cop who stops you that you have done a really good job tying this down. And in a 600 mile journey you will be stopped several times, not least: you are now in the bailiwick of a second tier of cops: commercial enforcement. They don't need reasonable suspicion, and as a matter of routine (for them) they will go over every mechanical feature of your vehicle with a fine tooth comb. Because that's what they do all day every day: random-stop trucks and look for defects in securement or mechanical.
If a police encounter is something that could go badly for you, doing something this attention-getting is out of the question. The laissez-faire attitude toward immigration law held in some sanctuary cities is most definitely not shared with rural police.
A mattress has no tiedown anchors. Handles won't cut it, don't even think about using handles as tiedowns! However the idea of carrying something you can't tie down is absurd.
To create tiedowns, wrap the mattress itself in rope, and this should be done in your bedroom so you have convenient handles to haul the mattress around. A mattress is both compressible and flexible, and it will really, really want to escape. The ropes will have many hundreds of pounds of force, from dynamic loading and your preloading the ropes in tension. (floppy-doppy ropes do nothing). So when you try to strap down, it will trapezoid. The only option, then, is to design the trapzoiding right into it and cinch it down hard so the whole structure is already in tension. It tends to not have a single square angle on it, and you're roping the mattress itself - that's why I call it "shibari".
If you want to plastic-bag the mattress under the shibari, that will make it even more slippery, which - well, that's probably a good thing because it will test the mettle of your ropework. If it's slipping all over the place while you're trying to haul it out to the car, you got an early warning so stop and fix it.
Your rope work needs to be tip-top. No granny knots, you need to use at least a proper bowline at every single intersection, preferably with something else as a slip-stop. Hope is not an operating doctrine.
Attach it to the vehicle
With the shibari providing a wealth of tiedown points, strapping it to the car should be a straightforward affair. Always bring the lines down at opposing angles, and in harmony with the shibari, so the ropes are in tension and everything else is in compression. Don't tie it square, or it will flop forward and back, and soon tear something apart. And expect everything which can move back and forth to do so.
Put as much static tension on it as is reasonable. The idea is that road loads are simply more of the tension you have already installed, not new tension in a new direction.
I would also consider a loop around the mattress and through the passenger door openings (with the doors open, that is... unless there'll be no one riding in that row of seats, then through the window is fine), "just in case" the luggage rack lets go. They're not really made for upforce.
I like Stan's idea of the 2x4's, but I would still have the shibari so you can bind the mattress proper, and strap down the 2x4's separately and also. If you wanted to stack other luggage above the mattress, I would put a sheet of plywood or OSB below the 2x4s, strapping all 3 things, the mattress proper, the ply/2x4, and the overtop items. Do not ask the top items to hold down the mattress, it won't work, the forces are much too extreme.
On the road, the mattress will tend to want to "nose up", the slipstream coming off your hood and windshield will try to lift the front and fling it off the car. I've moved mattresses this way many times, but usually below 35 mph and never over 50 mph, and monitoring the rope tension, it was well over 100 pounds of force even at that speed, and it scared me. Aerodynamic forces increase as at least the square of speed, so 75 mph is at least 225% of the forces I observed.
Unfortunately the typical goofus in this situation just throws on a few ropes and goes "la la la, drive like I normally do". Don't do that. Be super mindful and don't get complacent. On the other hand... if you take this thing down the state roads, you will get stopped by every small town local cop. So at least don't speed!
Also check your load after the first 5 and 30 miles, at every stop, and at the first sign of trouble. A stitch in time...