Map can export to ESRI, I will need to check further for the exact specification on the output however, you should be safe to continue with the proposal. And then we should chat further.
OK, with C3D 2009, which is also MAP3D 2009, the export formats are as follows:
AutoCAD Map 3D supports Arc/INFO version 7.2, 7.3, and 8.x, and E00.
AutoCAD Map 3D supports up to ArcView version 3.2 and 8.x.
You can export data from your DWG to a connected ESRI ArcSDE database.
In short it would sound like one might have a legitimate protest over this request for bids; as it seems to be targeting ONLY firms that have ARC-something or ESRI-other in it's most current iteration in order to comply with the request.
I believe that this particular practice is illegal in most states.
As this data format is only supported by those products. I can not speak to version compatibility with C3D2010 at this time.
You will need to verify the output version support of Global Mapper, as I see nothing specific in that regards on their web site.