With a good quality charger and/or inverter that can be more efficient. What you described is an online ups, lots of ham radio guys do it. And you don't need to be an electrical or electronic engineer to do it, you can buy basic off the shelf components.

Most of your inexpensive made for home use ups's are the offline type and when you try using a larger battery you have the issue of longer charge time and you could even run into an issue where the inverter is not designed to run longer then the original run time so it cold overheat.