If using Civil 3D you can export a LandXML of the Surface to AutoCAD C3D and then import that surface and apply a Surface Style to it that replicates the Colors of the Heat Map - my colleague Mike Carris is recording some videos of this and posting them on Trimble Library as a K-APP if you want to take a look to see how it is done. He will also show how to send 3D Faces to AutoCAD and how to render those in a similar way.
Here is the link to Trimble Library and the correct video
Click Here
if you cannot see the video (it may be because he has not made it widely available, please ping Mike on his email (michael_carris@trimble.com) and ask him to make it available to you. If you dont have an account on Trimble Library - go to this URL and set one up
https://trimble.retrieve.com/#/
You can use your Trimble ID to Log On - you can use the store to access Free Materials, purchase training libraries etc. and you can discuss with Mike directly other things that you can do with the Library as an individual or a corporation etc.
Hope this helps
Alan