I'm afraid I still don't understand your point. Perhaps I'm being stupid.
A broadcast transaction (be it standard of PoW) will not be accepted by the network if its ancestors contain a double spend. So any transaction fees the new transaction purported to claim will in fact remain unclaimed.
In this add-on proposal, I am not proposing that 50% of coins are generated via subsidy; rather, I am proposing that 100% of coins are generated via PoW.
Ok, 100% is fine.
Imagine that you are extending this branch: A -> B -> C -> D
My hashing power is higher, so I generate: N -> O -> P -> Q -> R -> S -> T
What will happen if N is a doublespending of A? Note that my branch becomes visible to you with some delay.