A different system. Why?
That different system would be a fork of bitcoin with 200 extra lines of code, or so, plus some verifiers which are just hash tables communicating with a simple http protocol.

The issue is this. The current bitcoin system has some freedom in what transaction a node works on, given a double spending attempt.
Right now, as I understand it, a node works on the first transaction it sees. Only after seeing a block with another transaction, will it change its mind.

Another possibility would be to flip a coin when a node has several colliding transactions.
Another possibility would be to ask a globally predefined node.
Another possibility would be to ask whoever the node feels like asking.

It doesn't matter; bitcoin works in any case. At some point a block will be generated with one of the transactions in all cases.

I want to use that freedom. The power of the proposal is that you have the ingoing transaction supply a name of a verifier.
There is nothing to lose by asking the proposed verifier, and there is a lot to win, namely ultrafast local trade.

By having the nodes put a transaction on hold, until either a timeout or a reply from the verifier, you can even have ultrafast local trade on Mars even though
most of the network is located on Earth. The general time loss by having the timeout is trivial, since it is similar to the latency in any case. On Earth it is within 200 milliseconds extra, say, in the worst case. The time delay is only large in exactly the cases where you want it. For instance, a Mars local coin, used in a vending machine on Earth. This is exactly the situation where suspicion is warranted.     

I hope someone understands what I am saying.

Cheers.