Don't panic. You are in better shape than you know.
The reason the water stops running in at the level it does, well below your basement floor, is that the ground-water level is below your basement floor. If you continually try to pump to the bottom of the pit as you were, you are trying to lower the ground water level around your house. That is a lot of water and it is continuously being replenished by water from the stream that is re-charging the ground water level. When you lower the ground-water level the rate of recharge increases
The effect is that, The more you pump, the more there is to pump.
You said it stops refilling when the basin is about 1/3 full, so it is probably 12" or so below your floor.
I would set the switch so that it starts pumping if the water rises one inch above the current no-inflow level. That has three effects. First and most important, it protects your basement from rising water. Second, if the pump starts running, that is a warning that the water is rising and you can check on the percentage of ON time to determine how fast the water is coming in. Third, it keeps the pump from running needlessly when the ground-water is not a threat to your basement.
If you install a backup pump, I would set that to start pumping if the water rises another inch above where the first one starts. That has two effects. First and most important, if the primary pump fails it pumps the water out to keep your basement dry. Second, it warns you that the the water is rising beyond the rate that the primary pump can keep up with.
Since you are selling the house, I would not put in a battery-powered backup pump unless you believe it is going to be worth that much to the buyer. The buyer can always do that if they want it. The price I saw on the Aquanot II was about $700, but it will probably cost $1000 to $1500 if you hire it done.
If it were me I would put in a second line-powered pump and a battery system with inverter, which I think I could do for about $500, but I am an engineer who likes to do such things. I also have a generator so the battery backup would only be necessary when nobody is at home.
I don't know what has changed, or if you are only now noticing that the pump operates more in the wet season. Maybe the pump level switch was adjusted or the pump was set lower. In any case, you have not mentioned that you have had a problem with water in the basement. Everything seems to be under control, and there is no apparent threat to your basement. Your pump has a capacity of 10 times the rate of inflow that you have been experiencing. Keep an eye on things and you will be able to deal with any issues that arise.