Xringer,
You can move a lot of water with a pump in a short amount of time. I don't know what the in flow of water your dealing with.
It sounds like your basement flooding is a random event and you need a toilet valve to turn on the ejection pump. You might rig up a light on the main floor letting you know the sump pump has been activated. A larger sump hole may allow you to run the pump once a day. Like I said I don't know what the in flow is. Once a day pump activation may be an interval you can deal with. Arizona is not a place where basements are common.
Sounds like you live on a leaky boat.
This may be a bigger problem then what meets the eye. Just a quess... your steam wall may be leaking. Steam wall - is the concrete seam where the wall and floor meet in the basement. You may need interior drainage system that will guide the water using drainage pipes to the well (sump hole) so that the pump can pump the water out as needed. If you have no warning when the basement is going to flood. The size of the sump hole you dig must meet the intervals you check the basemant for flooding... I'm sure get the idea.
Hope you can stay above water,
Vern