Sorry for the vague answer, maybe someone else can provide a better one with a concrete example.
If you've already got an Apps Script script, then this should be easy. What you probably want to do is implement a Calculated Model in order to get the results from BigQuery to your application. In short, a Calculated model will let you define a custom schema, and then write Apps Script code to query BigQuery and then create, fill, and return records (which will be very similar to the code you wrote to fill sheets). Once you've done this you can use the calculated model like you would a normal SQL or Drive Tables model.
You can read more here: https://developers.google.com/appmaker/models/calculated
and using this sample (although the use case for calculated models in this sample is quite different than what you're doing): https://developers.google.com/appmaker/samples/calculated-model/
Just like Apps Script you'll also need to enable the Big Query Advanced Service, you can find that in the application settings screen, all the way at the bottom, ("Advanced Services")