The lateral support on your bed mostly comes from the two connections at the top creating what is called a moment connection (think of lifting a long stick with both hands). The middle members do provide some additional support but consider that there is NOT one on the front and this should make it obvious that they are not absolutely necessary to keep the bed from falling over.
The middle brace is more about keeping the relatively thin and tall posts in place. In particular, the BOTTOM of the post. The braces are located in the middle for reasons such as aesthetics, to prevent toe stubbing (if the middle piece was not there then people would walk through and hit the bottom piece). They do provide some additional support against buckling but that is minor, again note that front piece does not exist.
My suggestion is to prevent the bottom of the posts from moving. If you have carpet then it's probably fine as it is as long as no one tries to move it. If hardwood floors or for the case of people relocating the post or bed, you can place a cross member in a different location, perhaps at one of the shelves so that it is not in the way, or at the bottom of the post, even under the posts, though you would need to rise the other end the same amount or cut down the posts in question