You can't get the campaign ID from the API response. However, you can use Facebook Analytics without using the Facebook SDK. You can log events by manually calling the API (which it looks like you're already doing), or through a Mobile Measurement Partner (MMP). As long as the logged event is associated with an advertiser ID or mobile cookie, the Facebook servers will attempt to match it to a user. If the matched user saw an ad in your campaign, then the install event will be associated with a campaign ID.
You can use Facebook Analytics to view information about users that installed your app from your campaign. See this article. That article focuses on using install sources in filters, but they can also be used in other analytics tools, such as breakdown tables.