Yes it is possible to have export functionality, i don't think it can be a right click operation but certainly you can add a button/drop down to export.
drop down right above pie chart.
There is an extension out in the branch where you can specifially target a objectid to export. may be you can create a Button in app and use that/place that button in your mashup.
You are right. I haven't seen a custom solution(mashup) where it has sense native controls you will get when you right click in Qlik sense app.
May be that is the way it supposed to work as we are embedding objects in HTML pages.
Hope this helps and if you are satisfied with my response please mark this thread with correct answer so that other people can find this useful.