It's an interesting idea. I will spend some time to think about it more and see if doing this without a hard fork is possible. You basically have to somehow create incentives for Litecoin miners to also keep track of the Bitcoin blockchain. It may be as simple as checking a few bitcoin block explorers, but why would miners want to do that for no benefit... unless they are forced to by a hard fork.
Wouldn't miners running older Litecoin clients (if this was implemented in a new Litecoin release) be relaying the wrong information and not keeping a complete and correct ledger for Bitcoin?
I wonder what the implications of miners/users would be who do not upgrade their clients. Assuming a fork isn't required, if a miner doesn't upgrade are they now mining on a different chain? If not, wouldn't their blocks they create not be complete as it would not have the Bitcoin information?
I can see upgraded clients rejecting the blocks created by miners using the older clients as they would be missing the Bitcoin information that is required to be stored in the block that was created.
You would do this with a miner vote. Basically implement this in the new clients. And have it in code such that when 95%+ of blocks mined in the last X blocks have this bitcoin hash data, then miners will enforce that every single block from then on must include bitcoin hash data. So this could take a long time (or never) but it will be relatively safe to do because when the fork happens, the majority of the miners are already running the latest code. And the minority would realize that they are on the wrong fork fairly quickly and would have to run the new code. Non-miners would be unaffected.
But like I've said, there's no good reason for miners to put in the extra effort to include bitcoin block data.