There would be bitcoin in a wallet that can only be accessed by a smart contract. That smart contract will integrate the bitcoin blockchain with another trusted blockchain provider. You need to trust the provider (the encryption, the code) the same way you trust the blockchain. How many here had a part in programming the Bitcoin blockchain?
Rather than paying the transaction fees for a much larger network, members can transact on a private blockchain and the network fees are paid as you move the value back to the Bitcoin blockchain.
I think your idea is very-close to or I would say exactly like - 'bitcoin sidechain'. Liquid from Blockstream is the best example of sidechain.
In Liquid network, user sends his bitcoin to the 11-of-15 multisig address held by the members of Liquid Federation. Once, his transaction has 102 confirmations, he got equivalent L-BTC on the network. He can do as much transactions as he wants on Liquid network at lower fees like you said. To get back BTC, user initiates peg-out process i.e. burn his L-BTC and request withdrawal to his BTC address. Functionary (a device held by Federation members) then verifies the output address of the user and issue the transaction.
To comment anything about your idea, I wanna know how your smart contract will work. On blockchains like ethereum, you get additional 'data' field so you can create smart contract like normal transaction without mentioning 'to' address and adding 'data' field. This will create contract address without the need of custody by anyone. But this is not possible on bitcoin blockchain. So, how would you create bitcoin contract address? Where will the keys of that address save? Who will keep custody of bitcoins received on contract address?
Assuming everything is secure with this process, what would your issues be?
In Liquid network, there is a backdoor through Timelock process. So if network is stalled for a time being, it is possible to move bitcoins with three emergency private keys. So, Liquid in true essence isn't 100% automated smart contract.
What is your solution for this? How will your contract handles payments and if consensus rule of your private blockchain doesn't meet for the period, will you keep a way-around like Liquid does?