It's pretty easy for an HR person to weed you out because of your degree. Talk to actual engineers you know in the industry. If they think you can cut it, I'm sure they'll be happy to pass your credentials and a recommendation on to a hiring manager.
The masters in CS can't hurt either; you just need to decide how you want to spend your time.
I agree. After you get your first job and get some experience you'll be good. It's just finding a way to get past HR. They just look at credentials and may ignore your unconventional interest in programming. Don't worry though, after that first job you are good. Recent experience and your reputation in the industry will outweigh your undergrad degree.
The CS degree won't hurt but I would only do it if you already had research in mind that you wanted to pursue. That seems to be the key to enjoying grad school - chasing your passion and interests.
The masters in CS can't hurt either; you just need to decide how you want to spend your time.