In CORE, one way to track proposals is to create estimates. In the Estimates screen, you can assign a win percentage or add any other field that you need via custom fields. You can send the estimate to your client and then later convert it into a budget, if required. Check CORE Help Center for details.
Another option would be to create a project for it. When creating a project for the proposal you are trying to win, use the Marketing contract type. This makes all your work on the project non-billable. If and when you win the job, you are allowed to switch it to any other contract type. On doing so, CORE prompts you to convert all non-billable time into billable time. This way you can track time, run reports and quickly turn it to a billable project.
If you buy the CORE CRM subscription, you can use the Quotes and Proposals features in it. Check Core Help Center for details.