To be honest I don't really have any clue and never worked with asp.net So I would not even know if it works like cgi or has built-in server capabilities.
I don't think it makes sense to add it to the LAMP app. But if so create a new app with asp.net?
Sorry, I didn't explain very well. Asp.net core is a new version of asp.net which does not require any internal ".net" / server capabilities. It is deployable to Linux, Mac, Windows. Basically the server app if fully self contained.