Assuming you only want to copy the assembly and its associated draft file, and none of the part files within the assembly, simply open the draft in RM. You will see the draft and the assembly below it. Select both, select copy, give them new new names or paths or whatever you need and then perform actions.
You can do the same with a drawing and a single part.
It gets a little more complicated beyond simple files, but all doable with RM.
I just tried with draft up and a sub assembly on the drawing. When I do save as I don't get any option to copy the sub-assembly. Whe draft is copied it still has original sub-assy in it - not a copied version.
It sounds like you are doing a save as from SE not using Revision Manager. In SE you will not have the option.