Sorry I don't have time to write a better answer. I would be willing to bet Nualart has thought about this problem at least and his answer could very well be encompassed in this paper: (In particular your problem might be a special case described in section 3)
P. Lei and D. Nualart: A decomposition of the bifractional Brownian motion and some applications. Statistics and Probability Letters 79, 619-624, 2009.