Is there any rule or loophole allowing a public referendum to override the senate to vote on a bill?

Many states have such a process.  But nothing like it on a federal level exists.

It's typical for elected officials to promise one thing then do another once in office (and exposed to the lobbyists, party pressure, etc.). Sometimes letter writing/call campaigns can change an official's vote. But often they don't care because our political parties tend to be able to protect those who vote the "right way" and manipulate elections to keep them in office. 

There is no rule or loophole allowing a national public referendum in the U.S., ever, under any circumstances, for any reason.  The U.S. cannot be overridden by a referendum.

In some states, a statewide public referendum to override the state senate is possible.

Many states have such a process.  But nothing like it on a federal level exists.

We don't have a national ballot initiative system, and I agree with you, it might be a better idea.

BUT here in California we do have an initiative, and it's badly misused.  Our state legislature is paralyzed by competing special interests, so if you want to get a law passed, you can 'buy' an initiative.

You need 300,000 signatures on a petition to get it on the ballot.  People are paid $1 per signature, and they set up tables in shopping malls, sometimes with 10 different petitions.  People sign them without really knowing what's in them.  Then when you get your signatures, you put ads on TV that lie about what the initiative does. And if you spend enough, it will get passed.  And bingo, you just bought yourself a law!

That would happen on a national basis too.  Just to look at our current president, you can see that a good percentage of Americans have no idea what they're voting for!

No. That's why we're a "Republic", and not a true Democracy.