Before electrical lighting, there was gas lighting. One type of gas operated light is the limelight, where a piece of quicklime is heated white hot in a flame of hydrogen burning with pure oxygen. Quoting wikipedia:
Limelight was first used for indoor stage illumination in the Covent Garden Theatre in London in 1837 and enjoyed widespread use in theatres around the world in the 1860s and 1870s. Limelights were employed to highlight solo performers in the same manner as modern followspots (spotlights). Limelight was replaced by electric arc lighting in the late 19th century.
In general, gas lighting was how all theaters were lighted in the middle of the 19th century. Gas lights on stage were controlled from the gas table, which was the contemporary equivalent of a switchboard.
By the 1850s, gas lighting in theatres had spread practically all over the United States and Europe. Some of the largest installations of gas lighting would be in large auditoriums, like the Theatre de Chatelet, built in 1862. In 1875, the new Paris Opera was constructed. “Its lighting system contained more than twenty-eight miles [45 km] of gas piping, and its gas table had no fewer than eighty-eight stopcocks, which controlled nine hundred and sixty gas jets.” The theatre that used the most gas lighting was the Astley’s Equestrian Amphitheatre in London. According to the Illustrated London News “Everywhere white and gold meets the eye, and about 200,000 gas jets add to the glittering effect of the auditorium … such a blaze of light and splendour has scarcely ever been witnessed, even in dreams.”
The main disadvantage of gas lighting was the fire risk, and once it was mature, electric lighting became cheaper than gas lighting mainly due to the lower fire insurance premium.
I guess this answer is too late for your performance, but it looks like either limelight or other gas lighting or else carbon arc lighting is the period appropriate choice. But beware that putting a real limelight in the theater where you are performing may get you in trouble with the fire department.