For dev development have the connection manager point to your non-prod data. For production values the project parameters is the preferred approach. You can set the values when you deploy to your prod database or, after deployment, you can go to the Integration Services node of the server hosting your SSIS packages and set the properties there. Alternatively you can write a T-SQL script to set the values directly in the SSIS catalog but I've not gone that route myself.
Solution configurations in VS are not the answer to this problem. They change how the project is built, not what is configured for it. These types of deployment changes should happen at deployment time.
I don't understand what you mean by not having access to SQL Agent. SSMS can be used to manually make the changes. If you're automating this then create an environment-specific params file (or have the deployment task calculate the values) and then deploy the env-specific param file instead.