Please note that " chaise longue de plage " isn't commonly used : we just say chaise longue. Lit de plage isn't used at all.
Un transat (masculine, pronounce the final "t") is the most French-sounding, it is an abbreviation that stands for " transatlantique ", that type of chairs being very often found on the decks of the transatlantic liners. Beware : unE transat (feminine, same pronunciation) refers to a transatlantic race with a sailboat.
Bain de soleil is also a very nice translation.