We’ve identified the cause of this issue. In some cases, when the Bridge loses Internet connection for an extended time, it may reboot and the local API can remain unavailable until the time is synchronized again. This will be addressed in the next Bridge firmware update.
Note that this does not explain cases where local access stops immediately after disconnecting from the Internet, such behavior can depend on router or network configuration (e.g. how local traffic or ARP tables are handled when WAN is down).