Can I drop and recreate a system extended stored procedure?
No, you can't drop and re-create xp_cmdshell. It's not even its own procedure, but part of a library in xpstar.dll (or at least it was at one point). And I am extremely skeptical that, even if you could re-create this procedure, that would fix the problem you're having.

Maybe instead of banging your head against this problem with xp_cmdshell, you should just bite the bullet and implement the functionality you need in CLR. Or perform the tasks you need outside of SQL Server altogether.