I tried with 2048 MB but it didn't work better.
The biggest 3D models in your project are Chopper, Darts and airplaine.

Is there any way to export my room into another program that perhaps handles memory a bit better than Sweet Home?

Use 3D view > Export to OBJ format... menu item and import the result in Blender for example.
Lights aren't exported. Please tell us whether it works better in other renderers or not.