I don't quite get what you mean by "distinct consecutive transversals", can you explain more clearly how the pre-coloring is done?
You write that you fix $\frac{(k+1)(k+2)}{2}$, for $k=\frac{n}{2}$, but for $n=8$ you only fix 10, and not 15. Do you actually want to fix $\frac{(k)(k+1)}{2}$ colors?
One question that I can answer: For the case $n=8$, with the precoloring you describe the completion you give is indeed unique. I checked by writing the corresponding boolean program and let a solver enumerate all solutions: there is only one.
