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It’s a requirement that your system as a whole isolates itself from the grid in the event of a grid failure. Most PV + battery installations are done simply as an adjunct to the household wiring – off to one side. A combined PV+battery controller could in theory cut itself off from your house and continue storing PV production in the battery until the battery was full but I’m not sure that any of them actually does that.
What is needed to allow things to continue working usefully is some means of disconnecting your entire house – PV, battery and everything else – from the grid. Some battery systems can do that, the obvious one being the Tesla Powerwall. With a Tesla Powerwall you have a Powerwall Gateway which actually sits between your house and the grid. In the event of a grid outage it disconnects your house from the grid entirely and will keep you going for as long as the battery and PV hold out. This however can only be done if you have something which provides that same functionality – disconnecting you from the grid.