Batch project input feature classes which do not reside in the default project folder become invalid when the subsequent inputs are dragged in from the map contents in ArcGIS Pro.
Make the added folder connection the default project folder. The error is unreproducible if the feature classes originate from the default project folder.
Instead of running the Batch Project tool, right-click the Project tool, and select Batch. Make a new batch tool and use it to batch project. The error is unreproducible when feature classes from non-default project folders are added as inputs.
Instead of dragging inputs in from the Table of Contents, browse to the file location to add them. The error is unreproducible when feature classes from non-default project folders are added as inputs.