How about a Privacy Web-Browser? For example IronFox is a privacy-oriented Firefox fork. By default it does send an user agent string. But you can make it install addons, for example an addon to change the user agent string to something generic (and fake).
To be honest, this is a very specific method, and I don't know which problem we're facing. So this might as well be some A-B problem. So what are you trying to achieve?