Hi,
You can try the below approach:
1. Create an EC on the account, EC is based on custom BO.
2. In the EC you will have a button to request account conversion approval.
3. On clicking the EC you can call the custom BO action - in UI designer on click of button trigger event. In the event call the custom BO action.
4. In the custom action you can set certain variables in the custom BO.
5. Create a standalone screen where authorized users can approve conversion requests - picked up from custom BO.
6. Once approved in step 5, set variables in the custom BO which will then display on account screen the status of the approval and conversion.