You would need either a complete MCX/Virtus upper to put on your existing lower or a non-Rattler BCG and MCX/Virtus barrel for your current Rattler upper. If you go the latter route, you’d need to find a Rattler-specific handguard that is long enough to cover more of the barrel.
The Rattler, [legacy] MCX, and Virtus chassis (“uppers”) have different formats and their respective handguards are not interchangeable across platforms. The Rattler BCG (recoil assembly, not the bolt) is different from the MCX/Virtus and would have to be swapped with barrel changes if you tried to use the same upper. *ETA: MCX and Virtus bolts are also different and should be used with their respective barrels; but if you buy a new barrel kit, it will come with the proper bolt.
Much of this is covered in the Rattler and Virtus threads in this sub-forum.
Also, Sig does not appear to offer a 10” MCX/Virtus barrel but does offer a 11.5. Sig sells individual Virtus components separately, so you could piece together a solution that works for your goals and budget, or you could buy a complete upper kit that will also come with everything you need to adapt the system to a standard AR lower.