B737 Captain reports being high and fast during CAT III ILS to Runway 16R using HGS. While attempting to intercept the glideslope the Tower issues a low altitude warning and the crew goes around.

Narrative

Vectored for ILS 16R by SEA Approach Control. Weather was low ceiling and limited visibility so we had briefed; and were executing an HGS [Head up Guidance System] Cat III approach. Aircraft was high on downwind and early base turn was directed by ATC. I had landing gear extended and slowed to extend flaps to twenty five. ATC queried several times if we would be able to 'get down.' Our response was that we would continue trying. After localizer intercept; we determined that we would likely be configured and stable no later then FAF 'OPVEE.' HGS CAT III mode was acquired and we continued approach past OPVEE expecting to acquire runway visually at 300 FT AGL. [At] approximately 800 FT AGL; SEA Tower issued us a low altitude warning and to immediately check our altitude. As we were still IMC; I commanded and executed a missed approach to sort out any anomaly at a safe altitude. All aircraft instruments and systems indicated normal; i.e. current altimeter setting; local frequencies; etc. No flags or aural warnings were activated on aircraft. We were vectored for; and executed a second CAT III approach to 16R and happily intercepted glide slope from below; achieving an uneventful landing on 16R. [This missed approach was caused by] ATC directing aircraft to stay too high; keep approach speeds too fast and turning base leg too soon during low weather approaches.