The marking should read "E. Lefaucheux/In-er Brevete", or "E. Lefaucheux, Inventor, Patented". The gun may have been owned or carried by a French officer, but as far as I can determine, it was not a French issue pistol. The French navy used pinfires, but they were made in French arsenals and none were of that exact type.
As far as I can determine, the gun is a civilian gun. Those revolvers came into common use in Europe in the early 1850's, when Eugene Lefaucheux improved his father, Casimir's, invention and applied it to a practical revolver. They remained in use until the 1880's when central fire cartridges proved superior. Some pinfire revolvers were converted to center fire cartridges.
As for the American Civil War, Lefaucheux revolvers were certainly used. The U.S. bought over 10,000 of the 12mm model, and they are recognized as a secondary U.S. martial revolver. The C.S. bought a few, but disliked them because they did not have the machinery to make the cartridges. One was presented to a fellow named named Thomas J. Jackson, whom I believe had some connection with the war.
Jim