if your missing anything at all, i would say its the concept of keeping the battery fully charged... if the current solar panel cant do it, you may need a bigger one.
this is assuming the battery is in acceptable condition, which i am questioning from the dialog that i have read concerning your issue.
because if your battery is in fact good and fully charged, when the pump kicks on, it will have plenty of energy in it to pump the water down and have lots of reserve left over when finished with its cycle....and assuming you dont have water running into the bilge unnecessarily from a hole somewhere, the solar panel will have adequate time to replenish the amps used from it on the last pump cycle....and be ready for next time.
with this said, I understand your focus on the seemingly odd behavior of float switch, but its a moot point if the battery is charged as it should be...
Now... you may ask my why the battery charge is more important than your idea of how the float switch should work in relation to how you see it working.....
my answer is,
#1.. because a battery with low voltage is extremely hard on any and all equipment that is trying to suck enough power out of it to run correctly.
#2.. because a discharged battery is not a dependable source for running a bilge pump
#3.. you obviously want the bilge pump to work all the time when it should, not just when you are around to help it function.
#4 if the battery stays charged, the way the float switch works when the battery is low is not and never will be an issue. problem solved!!!
dont overthink the symptoms, just solve the problem and the symptoms will go away