You don't seem to want to join the forum so you can see the guide to remove the handles but I'll put the button thing to rest here as you have confirmed the keycard is working by using the boot lock, pictures of the handle internals the button is not just as simple as you think squirting stuff round the side of the button may not be as effective as you might think unless the problem is a physical movement problem as the button is quite well sealed and not the button type shown earlier which would be for inside the keycard. It will most likely be the button on the handle as the RF receiver allows the door to be opened, the door panel inside will have to be striped so my advice would be get 2 new handles buy one first to make sure it's the problem.
Pros:
1. If your handles look worn they will be shinny new.
2. Less likely to need fixed again if you think you can clean up the button on the old handles.
3. If you sell the car the main feature of the locking not working will put doubt in the buyers mind and knock value off the car.
4. Sell your old handles on so someone can paint/wrap the outer shells without destroying the original ones.
Cons:
1. Buying new handles although there is an unpainted option I think around £15 cheaper and swap the internals.
Meganes do a repair kit which is just the internals but the button is not the same although the RF receiver could be moved across.
I'm sure all the help we have given so far is worth joining the forum if only to sell your old handles, or look up the guide.