LH has a very nice arrivals lounge in FRA for incoming F and C passengers (no Star Golds or other riff-raff are allowed in AFAIK - it is strictly for alighting LH premium passengers)
It is after immigration (a breeze) and offers many large and nicely outfitted showers with WCs and L'Occitaine products. There is a very comprehensive breakfast spread, with the obligitory sekt on hand for early-morning imbibers.
After experiencing this delightful way to break up a trip to the middle-east, there is yet even less reason to fly ST.