Would it be possible for a share sheet item to open a specific browser?
Yes you can. You will have to develop it yourself. Or the developers have to develop for their respective apps. I am a developer so this is an easy to do for me, but if your not a developer it may be more costly than its worth. Time, finances, hardware. You could look into jailbroken options though ( I don't know too much about that department )

If you are a developer look into the following :

UIActivityViewController

You will have to subclass it in order to customize it to your needs.

The proper documentation is located HERE