double-spends can be inserted in various places in the DAG that invalidate good transactions.
How can double spends invalidate other transactions? That would only be possible if the LCR physically discarded the entire other chain. I don't see why this is necessary; indeed doing this violates one of the tenants of efficient consensus 'never discard work'.
Don't forget a double-spend might not be identified until well after many derivative transactions depend on that spend. So you also have to kill all those innocent transactions too, which is death to a coin. You will think we can just wait for some window of time to be sure but...
Ok, I see what you're saying, but you don't need to discard all transactions that link back to the double spend; only the derivative transactions which spend the outputs of the double spent transaction become invalid themselves. I don't think this is any different than in bitcoin where you have a double spend with derivative transactions which gets orphaned... You can wait for the same amount of POW to go by in the transaction steam as you would have in N bitcoin confirmations.
You proposed a rule that selects the double-spend with the greatest succeeding cumulative proof-of-work, but it is possible this is also ambiguous. Can you really not graph scenarios to confirm this to yourself?
I am having trouble seeing how this is possible, you are correct. What do you mean by 'succeeding cumulative proof-of-work'?