You're right no one duplicates it and I suppose it cannot duplicate itself. In fact, the first thing that came to my mind was "echoed" because in the French sentence, what is seen in the water seems more independent from the letters on the barge than what is conveyed by "reflected".
It might also be interpreted as "reflected again and again"

An interesting question: I think 'reflected' is a more accurate description, because a reflection can happen on rough or smooth surfaces, as here, the surface of the water is obviously moving because the reflection is 'dancing and swirling'.
Whereas, 'mirrored' only happens when a surface is extremely smooth, like a glass mirror, or smooth water.
'duplicated in the water' would imply that somebody physically made a set of identical letters and placed them in the water.

Thanks Wendy. Could we have "mirrored in the water" or "duplicated in the water"? obviously the writer could have chosen "qui se réfléchissaient dans l'eau".

Hello Marina - No, I would use 'Its' as you have done. Ships are generally known as 'she', but I don't think one would generally refer to a barge as such - a barge is a workhorse, as lacks the femininity and elegance of a ship.
I would just suggest : Its name, La Reine, was written in large letters which were reflected in the water, dancing and swirling.
'tortillant' - twisting, turning, writhing, - all have a circular motion, whereas 'ripple' has a more horizontal motion as a small wave.

Would you use "she" , "her name" for a barge?