A block chain level integration of the two technologies is certainly very difficult with the current code base.
We would need an even more general and flexible and configurable crypto framework to build upon.
On the other hand it is possible to use exchanges or other trusted entities to transfer coins from one network to another. An exchange address is designated as a coin "sink" on one side. All coins received at that address are exchanged and then sent to the other network. This has already been done for coin relaunches. Here it would be a permanent mechanism to let the coins flow back and forth between the networks. Still, there has to be enough liquidity in the exchange to make it feasible.
In this construction the Cryptonote network would act as a giant coin mixer network and as a companion to the Luckycoin network.
I think it is complicated. Converting old coins to new coins on relaunch is one time job. Your solution requires some tusted exchange with liquidity all the time. Maybe the solution could be to create new coin - aLKY - anonymous LKY and use Proof of Burn concept. If someone from LKY blockchain sends his coins to 1 well defined public address in LKY blockchain without private key, he burns and destroys his coins. This will be detected by anonymous LKY client and it will create new address in anonymous blockchain with that exact number of coins. The new anonymous address can be specified in that burning transaction as commentary. Same can be done if someone wants to send coins from Anonymous LKY to old LKY blockchain. He can burn anonymous LKY and as a commentary will be new LKY address. This way, the only task is to implement coin burn detection on another blockchain. The drawback is, there will be 2 blockchains required for correct work. LKY will not be able to work without aLKY and vice versa. The cool part is,LKY blockchain will be same as BTC has, and aLKY will have same blockchain as Monero. Problem is, that blockgeneartion will require miners to add detect burned transactions from other blockchain. Mining can be hybrid, some on aLKY some on LKY.
Thanks for the hint on the proof of burn strategy. I never really looked into the details of proof of burn.
I have been sketching some designs of a flexible crypto framework that would allow to process different block chains.
In the end we might get to a point where we can feed different block chains into one wallet and fuse the block chains together.