I am currently working on my master thesis and I ran into this statistical problem. Hopefully one of you can help me, because so far I can only see that a mediation analysis with a MANCOVA isn't possible.
Yes, you can run such a model (with all variables included simultaneously) within the framework of path analysis using programs for structural equation modeling such as Mplus, lavaan (in R), and AMOS. In the free workshop below, I show how to run path analysis in Mplus: https://www.goquantfish.com/courses/path-analysis-with-mplus