I am currently writing a custom bundle and need to create a new Object Class for it. The class definition files however are generated in the base install var folder. Is there a way to place them in the bundle and have the base installation pick them up somehow?
no there is no way to do that.
You can install data object classes during bundle install though, see for example here.
To add to the previous answer, the idea here is to create a json file based on the class (you can find it in the var/classes folder), and move it to your bundle. Then use a command to import all the class definitions in that bundle. That’s the most proper way of installing data object classes during bundle install.