I think the cost to mines is the same as what's possible today, actually.

Consider a group of miners who wish to do this with no changes to the rule set. They can coordinate out of band and figure out if they have a majority of hashpower behind the decision to orphan a block, e.g. by signing a nonce with their coinbase keys. If they reach quorum, then they begin work on a parallel chain. Because they have majority they are guaranteed to eventually win, though depending on luck it may take a while. Because of this, assuming the external quorum system is public, the moment consensus is reached the other miners should all abandon the existing branch and start work on the parallel chain too, lest they waste work mining on a branch that is surely doomed.

The end result would be that the chain stops making progress, disrupting end users and generally creating uncertainty as the new chain is forged. Also, miners who built on top of the orphaned block end up being punished even if they did nothing wrong. Both these side effects are undesirable and unnecessary.

So the more I think about this scheme, the more it seems like a simple improvement on the current status quo. Miners can do what they could already do, but with a more reliable in-band signalling mechanism that doesn't require things like coinbase keys to be online, and them doing so does not disrupt existing users or waste energy.