You say that suspecting a bad key fob makes little sense since there is no leading indications it could be bad. You may very well be right about that. After going over your previous posts again I see there are indications the fob does work since you can lock the doors immediately after turning the car off.
You also say that there is no current draw on the battery that would indicate the memory is working while the ignition is off. If that is really true then you will need to find out why this is happening. Even if all the fuses are good and you have verified power is getting to them there could be a wire connection problem after the fuse panel. I suspect the memory fuse is in the panel under the hood since most, if not all, of those fuses are powered at all times. The fuse will most likely be around 10 amps and also power other things besides the memory. If you are going to try to solve this issue yourself you should invest in a factory service manual for the wiring of the car. It will show you how everything in the car is wired up. It is money well spent and will save you a lot of time in fixing this. Ebay is good place to get manuals for a reasonable price.
I had the same exact issue the car you have to press lock as soon as you get out otherwise the system will not lock, with car on all locks work fine but as soon as I turned it off no buttons on doors would work and the key fob I had to press within 30 seconds otherwise I would have to lock all doors manually. But After 3 months of trying everything every one else told me it ended up being 1 fuse. I have a 2007 Lexus IS 250 see photo for fuse it’s located under the hood on the passenger side.
FOB door switch circuity works, but only works for 30 seconds? hmmm … Definitely the sort of frustrating computerized car problem that seems to be getting more and more common by reports here. It would be helpful to others to get to the bottom of the issue. Do you know what that fuse is for? There’s often a diagram or chart in the area of the fuse box , often on the underside of the fuse box lid, that gives a name for each fuse. Did you take this car to a shop for a diagnosis?
OP & @salinarios_187586 , did you notice any other problems besides the key fob? Did the headlights work? The other exterior lighting? The horn? A/C? Do you know why the fuse installed seems to be a 20 amp version, when it should be 10 according to the chart above?
In any event, suggest to inspect all the car’s fuses. A fuse that is too low might cause something to not work, but a fuse that is too high could cause a car fire.
