You probably need a bigger sump.
You don't want a pump that doesn't use electricity. Electric (off the grid) is the most efficient.
You need to apply some engineering.
There is nothing that is described below that you can't do for yourself. Your plumber should have been able to solve this problem for you. He has not served you well. He is apparently selling pumps.
Find the outlet and measure how much water you are pumping out in ten minutes. Bucket and watch with second hand (they still make them). How many gallons per cycle? How many cycles per hour. Make sure the water is getting out where it will not drain back. Measure at least 5 pumping cycles.
Then measure what is the difference in water level in the sump between when the pump comes on and shuts off. Some pumps have very low range switches and you probably need a separate switch. I would have two switches, an alarm, and operate the pump (or pumps) through electrical contactors.
Given what you say about your demand, I would have two pumps operating with grid power, with an alternating duplex controller (I build my own controllers with a commercial alternating relay) and alarm system, and a generator to supply power if the grid is out.
If you want a battery backup to operate while you get the generator started, that's fine, but I would not rely on it for extended outages.
Find the model and rating of your Zoeller pump. Compare its GPM at 15 ft of head with the GPM that you are pumping out.
Now, you want to put in a sump that is big enough and deep enough to give you at least a 10 minute cycle.
You want a pump that will pump out the water you get in flood times. If your flood time incoming is 20 gallons per minute, then you should have at least a 40 GPM pump, or two pumps that total 50 GPM.
Your pump capacity should be at least 10 times your baseline (dry season) flow rate.
With serious flow problems at flood time, I would want two pumps that run off the grid. Battery powered pumps will not do the job for long periods. Compare a possible battery system with your flood season demand. Will it keep you dry while you get a generator running? If not, it is no good.
Don't count on buying that generator after the power has been out for 6 hours and your batteries are dying. Lots of other people got to HD ahead of you.
Now all of this may be overkill. If your problem is no check valve, or a poorly set switch, or too small a sump, then you may be able to get by with your one pump.
If the problem is that you really have a lot of water to pump, then you need a robust system that will let you sleep at night, or take a vacation during the rainy season, without worrying about whether your basement is going to remain dry.