What I think you need is called a "fanout" in FME.
A fanout creates separate outputs depending on the value of an attribute or other value. There are two types of fanout. The Feature Type Fanout would create a separate layer in a DGN file, for each value of the attribute. The Dataset Fanout would create a separate DGN file, for each value of the attribute. You need a dataset fanout.
In the Navigator window in FME, check the parameters for the DGN writer. You'll see a parameter called Fanout Dataset. Double-click that and set the parameters, like so:
In this example I am converting a Shape dataset of cycle paths to DGN, and creating a separate DGN file for each different value of PathName.
You have to make sure you include the file extension in the Fanout Expression, as FME won't add it automatically.
When run, this is what I get for the output:
I hope that is what you are looking for. If not, please do respond and let us know.