I don't think this is available for Fusion 360 (yet).
There is no other CAD software which can control Fusion 360 models! There are however tons of programs which can run the geometry model that Fusion 360 uses, maybe that is enough for your case? Two options that come to mind are AutoCAD and Inventor which can both run headless.
In the future Autodesk might add Fusion 360 support to Forge Design Automation. Fusion's 'Big Brother' is already there: Autodesk Inventor. I feel like this might be more likely than Autodesk creating a headless version of the tool.
If you are not too deeply invested in Fusion 360 and you really need this functionality than Inventor is a good choice. It is very similar to Fusion 360, including it's API.
As for your reference to SolidWorks, Inventor is it's closest competitor but it is created by Autodesk just like Fusion. This generally allows for better compatibility.