So in your case, the data source is "get date"->web and you are trying to send requests to the same web api but with different parameters? May I know how do you "pass in the querystring of the web api url", AFAIK, the dataset for the reports that published in Power BI Service doesn't accept any parameter.
By the way, I see you mentioning embed in web page, in case you would have any interest, check How to Embed and Filter a Power BI Report with the new Power BI Embedded JavaScript API.