It might be possible to task organize the units, but allow the men to rotate jobs once in a while by swapping out weapons when there's a quiet moment. Or would that cause too many complications?
That could work. It would sacrifice some of the specialization, since it would make the designations strictly TO&E based, not soldier based.

If you organized permanently this way at what level should it happen. Companies with support, security, and assault platoons, or battalions with support, security, and assault companies?
Definitely not at platoon, thats all I know. Missions and situations change too much and too rapidly to make specialization at platoon level feasible. Having a weapons squad is enough. Flexibility in the "short sword warfighting" is too important to sacrifice.