To me, publishing is intended as a non-interactive activity. You should be able to start the publishing process, go get a cup of your favorite beverage, and come back to find the publishing process complete.
Calling input is absolutely an interactive activity.
I would extract the interactive section of your code that calls input into an interface function and turn the computational section of your code into a computational function that accepts data either from the interface function or directly from the user calling it. Then call publish with the 'codeToEvaluate' name-value pair argument containing a call to your computational function with the appropriate inputs. If you want to have a user interact with the computational function they could do so directly or they could call your interface function.
If you're publishing via the Toolstrip rather than executing the publish function, click the downward pointing triangle under the Publish icon on the Publish tab of the Toolstrip in the Editor. You can specify the code you want to run when you publish.
