For the smart locking to work, that 3 button key will need to be near the door.
As you walk away the doors should lock.
As far as I am aware (see below), those small buttons you refer to are receivers; they are not push buttons (but I could be wrong on this as I have never tried to use them).
There is a larger button on the lower edge of the tailgate, inside, which should function to close the hatch.
I have the same keys as you do, but only use the 4 button as my daily key.
Indeed, my daily set do not even have the four keys, or internals, present.
I used the smart key for locking and unlocking.
You do have the drivers door 'lock' switch enabled ?
Of course, the 4 key unit is used in the ignition as it contains the immobiliser key.