There may very well be a problem with the networking components on the PC that is preventing NET USE from working properly, but assuming that the system itself is functioning, this is the formatting I would use in the batch file:
net use p: \\10.200.1.1\sys
copy *.* (or whatever files you need)
net use p: /delete /y
net use p: \\10.200.2.1\sys
copy *.*
net use p: /delete /y
net use p: \\10.200.3.1\sys
copy *.*
net use p: /delete /y
etc.
You don't need to specify a "persistent" parameter at all unless you specifically want a drive mapping to be persistent. The /delete /y parameters will force Windows to delete the drive mapping. If you don't include the /y parameter, Windows sometimes will not actually delete the mapping since it is waiting for you to answer yes or no to its prompt of whether or not you want to delete the mapping.
You can test the system's functionality by manually typing these commands into a Command Prompt one at a time. If all commands work manually, then they should also work in a batch file. If they do not work, then the error you get (or the command that fails) will help us know where to look for the problem within the OS.