Multivariate Student-t Posterior Predictive - Detailed Derivation

I'm a CS Ph.D. student wishing to improve my math skills by solving problems relevant to my research. I started this derivation of student-t posterior predictive some time ago and I keep getting stuck on it. I think I managed to go very far already, but I'm running out of time.
In the current exercise, I'm tackling a multi-variate posterior predictive with unknown mean and covariance, and multivariate outputs. I provide my notes with several pages of context and then towards the end you can find a spot I got stuck in. The line's unfinished, I tried "approach 2", some intermediate solutions on a side, but nothing works.
I know the solution is multivariate student's t and the problem is almost finished. But due to my background, I'm looking for someone who would locate potential mistakes and finish the solution in an extremely detailed way, preferably not taking massive shortcuts due to usage of "trace" and "determinant". I think the main issue I have is I constantly need to look up all the matrix operations to make sure what I do is kosher, and I'm probably missing some identities. For example, someone suggested a Woddbury formula.
So to be clear again - I'm offering a higher bounty primarily for the learning experience - not for the final solution. And I'm open to negotiate the price based on initial proposal.
Also my writing is horrible - don't hesitate to ask for clarifications!
Thanks!