This sounds pretty much like lightning, with the important difference that the tokens are not transferred off-chain.
They are transferred off-chain, if by saying off-chain, you mean "outside the main Bitcoin network". But yes, they are on-chain, if you define on-chain as "on any chain, for example sidechain".
In general, if you imagine a network, where all LN transactions are shared with every node, then you will get my idea of decentralized sidechains. Because, according to the rules "sign your coins to peg them in". And LN closing channel transactions contain valid signatures, so they can be pegged into such sidechain. It is even compatible with hiding internal data from third parties, because penalty transactions are encrypted, and you only share them in encrypted way, so the network can see basic things, like "this UTXO was already used", without having direct access to that data, and without having an option to close the channel, if the creators are not willing to do so.
Which means, that the whole product of LN can be used as an input for the sidechain. And the output would contain just some batched transaction, broadcasted into the main Bitcoin network.
"Bitcoins" in a sidechain are tokens issued, the Bitcoins in Lightning are actual UTXOs that have not been settled in the Bitcoin blockchain yet.
You don't have to "issue" a different token. You can just reuse, what is already there. In the same way, you don't have to invent new signed transactions to peg coins in. You can reuse existing output, produced for example by LN (but it is not limited to this case). And then, if you reuse existing signed transactions, then there are two options: both networks are IOU, or none of them are IOU. Because you can use exactly the same bytes in both, point at the same UTXOs, broadcast the same transactions, and so on.