Are aircraft separately certified for CAT III ops for +2500 feet elevation?
Yes. In certain jurisdictions.

Such as Europe, if it has not been demonstrated during flight testing.

This is covered in CS-AWOS / Subpart 1 (which covers Automatic Landing Systems).


  In accordance with CS–AWO 132, the effects of aerodrome conditions (e.g. elevation, ambient temperature, runway slope and ground profile under the approach path) are to be investigated and, if necessary, appropriate limitations derived for inclusion in the aeroplane Flight Manual. Guidance is given in paragraph 5.


Paragraph 5:


  Elevation and Temperature. The effects of aerodrome elevation and ambient temperature should be examined where operation is envisaged at aerodromes about 750 m (2 500 ft) or in temperatures greater than ISA + 15°C.


A performance demonstration would then be required (but not additional equipment).