Chris, I use this method constantly in my surveys.
Sometimes it is just like your example where I need a channel type variable. At times, it could be a sample source variable (e.g. panel sample vs client sample).
So adding "channel" to the end of your survey link as a pass-in field is just fine. Define "channel" as a pass-in field and set the data type to whole number as you indicated.
Ensure the correct channel is applied to the appropriate survey link.
No need for a free format question. As long as you define "channel" as a pass-in field, that will suffice.
Click on your 4 survey links for each channel and ensure they all work correctly. Check the data too.
If you are using unique IDs (or passwords), as an alternative, you can import all IDs / passwords and channel sample in and define "channel" as a merged field. In this situation, you would setup a CSV file with 2 columns (ID / password and channel). Once you have set your sample up, import it into Sawtooth Software. You can also check the sample has imported okay by clicking on the "Password Data" button (click the Access button at the top, select the Merged Fields tab to get there). Also click on the red tick button to check all IDs / passwords are unique. You don't need to add the "&channel=X" to the survey link when using this option. You can just use the same survey link format with the only difference being the unique ID / password.