Blockchain Network Communication Doubt In wallet application

I am trying to understand working of ethereum based wallet applications. Suppose I have to build a wallet application called MYWallet that supports sending and receiving BTC.
Users can send and receive BTC either from a wallet address created within the wallet(internal wallet) or from any external wallet.
If a user for the first time adds 5 BTC to his MYWallet BTC address. This transaction comes with the bitcoin network knowing about it and then reaches the MYWallet address from the external wallet address.
If he then sends 1 BTC to a different internal wallet of another user within MYWallet, does the transaction have to be transmitted to the Bitcoin network?
Within the MYWallet app, the 1BTC can be subtracted from one user’s wallet and added to the target user’s wallet easily.
But, how does the network know about? When is the best time to transmit this data to the bitcoin network?
Is it ok if the internal transactions between users in the wallet don’t transmit to the network?