What the author wants exists, just not by the names that the author is likely searching for. Victron makes combination charger-inverters that takes power from

What the author wants exists, just not by the names that the author is likely searching for. Victron makes combination charger-inverters that takes power from the grid and keeps separate batteries topped off and also supplies AC power via inverter. They sell these as their 'Multigrid' and 'MultiPlus-II' products and are populat setups for boats, where you have shore power but need to be able to unplug the shore power at any time and keep things going. It's also a feature of quite a few cheap hybrid solar inverters now- they can charge batteries from mains if there is no PV input. These work as a UPS but you do constantly run the load thru the inverter, thus your are taking a small efficiency hit; however it is truly uninterruptible. There is no 'switchover' time, which is also what makes a lot of UPS hardware so specialized- many are doing mains switching in essentially 1 cycle of 50/60hz mains power. Theres all sorts of stuff that goes with that. Meanwhile, charger-inverter combos ditch having to deal with that by just going AC-DC to battery and then DC-AC. You always have clean uninterrupted power from the inverter. This used to fuel reliability concerns but these days much of this hardware is in applications running 24/7 for years now.