I have three different instances of SDE that have the same schema. Each SDE has multiple (5) feature datasets. I need to reproject all feature classe within these datasets to a common coordinate system (in this case web aux sphere), merge the three SDE together into that coordinate system while maintaining all the feature classes, and overwrite old data in the previously merged "final" database (which is just a file geodatabase not SDE). I haven't been able to figure out how to automate this process. I've tried using an iterator but my ModelBuilder and ArcPy experience is limited so I have had no success.
The figure below may help answer your question. I would create a new file geodatabase, with your desired schema, and would set the output from Project tools to be saved there. If you need your final database to be an SDE, then maybe you can use one of your original SDEs to store your new, projected data.