Current code leans to GFCI the heck out of it, and if it nusiance trips or fails (as they do with some regularity; typically about 10 years) and backs up sewage or water, tough noogies. People who sell GFCIs write the codes, and they have been expanding places where GFCIs are required for years. As of 2002 code (at least) there was still an exception for sump pumps and freezers, but those have been removed due to either "greater care for your safety" or "vast sums of money from GFCI makers .vs. no sums of money from people with flooded basements and freezers full of rotting food" in the code-writing cycles. [It should be self-evident that the linked page is from the side with all the profit in this game, and just a tad smug/biased.]
I would NOT change the ejector pump to a dual, nor add anything to its circuit - sewage backups stink. A dedicated breaker with one outlet is one way to be certain that nothing OTHER than a pump failure or power outage causes the ejector pump to stop; and depending on the pump, it may need the entire circuit to itself, period. Even if it does not, sharing is poor practice, IMHO. If you would like to add a GFCI to it, put in a blank face GFCI in-line before the outlet, or a GFCI breaker. If it's installed to code as of the time it was installed, I'd suggest leaving it alone, but that's up to you.
In most cases an extra sump pit is pure plumber profit - if the sump is not tiny, the backup pump can use the same sump, and just be set to come on at a higher water level.
The practical problem with your vision of "a battery with a 110V outlet" (battery charger, battery, and inverter) is that is entirely possible, but wickedly expensive; also, batteries don't last forever, both in the "length of power outage" and the "number of years before you have to scrap them because they no longer work at all" sense, so it gets even more expensive. Sump pumps with a battery backup built in are made - they have many of the same problems, but usually skip the added inefficiency of converting back to 120 VAC and just use a DC pump.
If you have utility water, and such things are acceptable to your utility, the water-powered sump pump has the benefit of being fairly inexpensive to own, being based on a power source you don't have to buy and maintain. You will get a big water bill if you need to use it. If you have a well pump, it's useless.
Anything depending on you running a generator depends on you being home, the generator being functional, and adequate fuel for the generator being available in the event of a prolonged outage - which it often is not, since the filling station does not have power either. You can expensively address most of those with an auto-start natural gas powered unit if you have natural gas available...
You can also use only tile or concrete for surfaces in your basement and not put things that would be bothered by getting wet in it, so that the occasional flood is less traumatic. How badly or quickly it floods if you turn the sump pump off might guide your thinking on this. It can be very expensive to build and maintain a pumping system that will work though a long, widespread power outage.
Best Answer
Another answer by Fresh Codemonger suggests having two pumps, one on regular power and the other on your inverter.
I would like to suggest a similar option that I think is more efficient.
As the other answer says, you should have a second pump. However, instead of connecting it to your inverter, your second pump should be one that is designed to run directly from 12VDC and have a battery charge controller. That way, the pump itself maintains the battery charge and automatically turns on when needed. That removes the need for the inverter, resulting in a simpler and more energy-efficient setup.
There are also a number of sump pump companies that actually make a 2-in-1 pump which has both a 120VAC pump and a 12VDC pump in the same unit. It automatically switches between them as needed.