There is a third option.
We know from the discussions about dueling in WoK that it is possible for a Shardbearer to lend his or her shards to another person (don't have the book handy to give a direct citation, but it's mentioned that a duelist can ask to borrow the king's plate and blade somewhere in there). There's nothing stopping the Azish government from having specific bureaucrats who are bonded to all of their shards, who then lend out the shards as need be.
Naturally there's a lot of questions for how exactly it would work. We don't know if lending shards requires close proximity, or if it involves unbonding the blade from the original owner and bonding it to the other, or even if it's just an Alethi cultural thing. But it does seem like a far more secure option than "lock them up in rooms".