2011 D4D T-Spirit "key not detected"

[Hymerist]

Hoping someone can help.

A recent vehicle Battery fail and subsequent new Battery fit has caused the fobs to not be recognised with a "key not detected" error.  Despite knowing them to be good I replaced the fob batteries to no avail.  The curious thing is that if driven enough (say three short journeys in a day), the fault appears to clear and both the smart entry and press start all work (for a few hours) left longer and the fault returns.  £100 at Toyota dealership achieved nothing other than "we don't know what's wrong with it".  They did muse that maybe replacing the smart entry ECU might fix it but at £800 I turned that offer down.  I have checked and cleaned all the fuses and thanks to the Dealership, the fob batteries were replaced again.  the fact that if used enough the smart entry begins to work as normal seems to indicate nothing "blown", almost as if some power source for the smart entry is charging with use and depleting over time?  Any thoughts would be welcome as I can't afford to keep throwing money at it but it is making the car difficult to use for my wife..