You can buy "automatic" or "submersible" bilge pumps that have float switches built in. Those are more common than those with external float switches and are extremely reliable.
They're $25 up on eBay depending on the size you buy.
BTW it's typical for boats to have a solar trickle charger to keep their battery banks topped off. They are small and relatively inexpensive, but the cable run to a basement probably wouldn't work well - DC voltage doesn't hold up with long cable runs.
You can buy a "maintenance charger" to keep the battery topped off for $6 [on sale at HF] to about $20.
500GPH is about the smallest and don't forget that volume would be reduced with a high head. The blue base comes off and there is a stainless steel mesh strainer underneath. Sump pumps are generally more like 2-3,000 gph.