I fly the 737 NG across Europe. A simple solution to prevent this from happening again is this...
We cannot approach an airport with a 3800 ft runway, because such an airport is not loaded in our FMS nav database. Therefore the TAWS will start yelling TOO LOW TERRAIN when below a certain radio altitude ( unless terrain inhibit is selected)
You might want to dig deeper into your system details. You might be right given what database option your company installed, but at my US carrier and it sounds the same at SouthWest, any airport with a runway >3500' in in the GPWS database and will not sound off if on normal approach course.
This is different from the FMS database where the airport options are much more restricted to approved airports available to display on the screens.
Me thinks it's that way in case of a split-s whilst on fire and landing on the first available piece of concrete. Last thing needed would be a "Too Low Terrain" blaring.