I do not think that, without the LNs making profound structural changes in their design, the Safe Network can manage that kind of transactions.
I think it would be less difficult to create an altcoin (SafeBTC), inside the network, pegged to bitcoin in a 1:1 ratio.
The user would exchange bitcoins for SafeBTC that would be added to their wallet allowing transactions to any user safely and quickly with minimal cost. When the user decides, these SafeBTC are reconverted into BTC again by adding a transaction to the BTC blockchain.
The big difficulty is how the bitcoin<->SafeBTC conversion is implemented and who store and manages the private keys (possibly a multisign of several different groups).