What is the problem with a sump pump?

Discharging sump pits into the sewage system is not allowed in most places, since it adds to the sewage treatment plant load.

Installation would be very challenging- it would need to go under my basement stairs in a pretty tight spot. Also, I believe that there's rock under my basement slab.

The floor drain does not flow into a public sewer system or into my septic system.

Thanks!

I have what could be a similar problem in my garage.
I had to core into the ledge to get the foundation in which lead to water seeping and a difficult way of coring another hole to put a sump pump in.

I do have a utility pump placed in a cut down drywall bucket (could only get so deep) and set it up on a timer.
What I learned was I only needed it to run for less than a minute to get the garden hose running and it siphons the balance.

Don't know if you have a lower point in your yard near by to do the same.

Works for me, hopefully can work for you.