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.