No, you don't need to do anything.

If so then in that Case I assume that the accurate balance in the Cold Wallet does not need to be reflected as it is only used for spending?

Correct. Without a means to gain knowledge about received money, the cold storage wallet can't update its balance. But it holds the private keys required to spend the money it received. The watch-only wallet can be used to check the balance without it having the ability to spend any money.