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Contrary to popular belief, I think it is unlikely that the family name Lax was based upon the German word lachs. Lachs in German means salmon.
I expect that, as with the family name Katz (which in German is a cat), the Jewish people could quietly make fun of the Germans. The Germans would think it was a "funny" name, having a family name based on a fish or animal, whilst the joke was really upon them for not appreciating the true origin of the name.
The most likely origin of the family name Lachs is from the Lachs speaking region in North East Moravia - which comprises parts of Czech Silesia and the Hlučín Region, (around Ratibor in Silesia (then Germany and now Poland)) .
The Czech composer Leoš Janáček wrote a suite "The Lachian Dances" based on folk songs from that area. (His work Taras Bulba is particularly well known).
Many people still speak and write in the Lachs language.
The spelling of family name Lax rather than Lachs was in use on documents before 1812. A potential reason for the different spelling would be to comply with the Silesian law of 7th May 1791: to have a name which sounded the same (and could be written the same way in Hebrew script) but was not identical (gleichlautent) in German spelling. This would enable a younger son to continue the family name whilst complying with the letter of the law.
Peter Lax