There were indeed discussions about four years ago that LH will buy the name rights for "Swissair", but I don't know the outcome of that. That deal would also make sense for debaring the competitors from buying this brand.
Concerning the designator, the ICAO-code SWR (mainly used by the air controllers) was re-introduced in October 2002 and replaced the former Crossair-Code CRX.
But honestly, for most foreign passengers, SWISS sounds like a rebranding of Swissair. And I know the history and can live with - my Swissair-memories contain mostly the dreadful black biz-seats on the MD11, which were by far worse than nowaday's luxury-biz on the A330-200
And don't forget that the brand of SWISS's new sister-company is also just "AUSTRIAN"...