By the addition of entropic regularization, multimarginal optimal transport problems can be trans-formed into tensor scaling problems, which can be solved numerically using the multimarginal Sinkhorn algorithm. The main computational bottleneck of this algorithm is the repeated eval-uation of marginals. Recently, it has been suggested that this evaluation can be accelerated when the application features an underlying graphical model. In this work, we accelerate the computation further by combining the tensor network dual of the graphical model with additional low-rank ap-proximations. We provide an example for the color transfer between several images, in which these additional low-rank approximations save more than 96\% of the computation time.