This sounds awfully complicated, near impossible. You'd need a circuit that disconnects from the grid once it detects a brown out or black out, then produces a near sine wave all by itself, then re-syncs with the grid once it comes back up and transfers back. And all that has to happen cleanly enough that even a computer connected to it doesn't glitch, truly a "delicate maneuver". No way this can be produced for less than a couple thousands of dollars!
It's called a line-interactive UPS. You can buy it for $200.
Line-interactive gets around it by simply using a inverter for both grid power and battery power, thusly being able to produce it's own sine wave inside the house. You can also use an VFI those convert grid to DC and then plug the battery in there, then convert back. Same effect.
A line interactive UPS does not sync necessarily sync you to the grid, though the electronic will usually try to keep it in sync. It's not necessary here.
It's called a line-interactive UPS. You can buy it for $200.