If the layer that is available in the other channel, with the known coordinate system, has the same vertex coordinates in both sources, you could try using a CoordinateSystemSetter transformer to "tag" all the dwg data with the known coordinate system. Reprojection wouldn't work here as that process requires the dwg data to already have a coordinate system assigned.

However if the known coordinate system is not the "local" system you suspect the dwg to be in there are only a few options available to you (that I can currently think of!)

1. Get the local coordinate system details from the DWG author (usually not possible, or too hard) and reproject
2. Simple: Attempt a bulk shift of the dwg data to match the two corresponding layers (try the Offsetter transformer)
3. More complex: Attempt an affine shift of the data (use the AffineWarper transformer) using the two corresponding layers to construct control vectors that define the change from "Observed" (dwg) location to the controlled (known coordinate system) location

Hope this post points you in the right direction.