Stand by for the briefest of history lessons. La Dauphine was the name given to the wife of the eldest son (the Dauphin) of the regnant King of France. In this case, the Dauphin was another Louis, son of Louis XV, who lived from 1729-1765. His first wife Marie Teresa of Spain gave birth to a daughter then promptly died. This regiment was raised in 1747 in her honour and, for a time, would have had her coat of arms placed on the flags. After her death, Louis then married the Saxon Princess Maria Josepha and her arms are in the first picture and on the flags.
She was to give birth to three Kings of France (not to mention a host of daughters) Louis XVI, Louis XVIII and Charles X.
The regiment was probably recruited in Saxony. It disappeared in 1760 when it was amalgamated with another German regiment.
The figures are by Crusader except for a Foundry officer and an Elite standard bearer (the left hand one as you look at them).