Hello and welcome to the LOC.
How did the locksmith guy test the battery, mine was doing the same with the light flashing but no operation and a new battery sorted it so I would for the sake a couple of pounds fit a new one before anything else.
Not sure if this carried over from the LS400 but to reset those fobs you simply put the key in the ignition, turned to ign position and left it for around 10 minutes,remove key, reinsert and then it was reset.