Sunsynk will give you what you want. You can define a minimum battery level, after which it'll switch to grid if there is not enough solar to supply the load. If there is no grid then It'll continue to use battery until a final emergency cut-off that shuts the whole system down.
The battery setting can be configured on a timer, so you can have different cut-offs for dulifferent times of the day.
Can't second this enough. I have a Sunsynk inverter here in South Africa where we deal with multiple-daily grid outages. So the priority is always set (based on time of day) to keep the batteries charged "just in case". If Solar can't provide enough juice, it sucks from the grid.
The whole thing is actually quite elegant. It turns "electricity" into this fungible thing like it was supposed to be in the first place. More pro points, you can plug in an electric-start generator and configure it to start automatically as needed in order to supplement the solar. Going further still, you can plug in a wind-turbine and have it blend it into this whole system.
Makes me wish I could install a wind turbine on my roof. A fair bit of solar, coupled with a little bit of wind suddenly makes your batteries go a really long way towards being fully off-grid capable.
The battery setting can be configured on a timer, so you can have different cut-offs for dulifferent times of the day.