That's a little more hard to answer of course, the only thing I can suggest you do is look through the record of the EU Parliaments version of the Hansard and see how British MEPs have voted for a direct look at how our MPs have influenced voting. Of course, thats a pretty dull task.
Not enough time in the day for that

I just don't buy the "we'll lose our influence argument". The Norway, Swiss, Icelandic option is not perfect but as below, their people don't seem to mind:
It’s been many years since any opinion poll showed a pro-EU majority in an EFTA nation. Iceland has formally withdrawn its application, and the Swiss pro-EU campaign has admitted defeat and closed down. As for Norway, the latest survey there showed 17.8 per cent in favour of joining, 70.5 per cent against.