What's wrong with the model as you have it? Do outer (left) joins from the fact table to the dimension tables.
If you have two Fact Tables and there isn't a one to one relationship, you may have to have several connections in order to achieve the right level of detail for what you are attempting to visualize (or you'll have to use aggregates and window calcs to attempt to fix it). IMO, since you have data creation access, the approach I'd take is to create views within your datasource that align the data for the level of detail you expect to see in the viz. It might mean that you have a few views -- then link them together using action filters in a dashboard -- navigating to more detail, but with the action, fewer records returned.