Some pumps have optional extras available for problems like yours - like a pneumatic (air) operated remote button to get the pump to kick in. I've seen DIY installations with a wired push button connected across the flow switch - though that might not be safe in a bathroom.
Or - if you've...
The float valve must have a fault that's letting the cistern overfill, raising the height of the overflow isn't really a solution.
You've need to see if you can get a new washer / diaphragm for the float valve, or just change the valve for a new one.