One of our branches wants to run a sweepstakes, whereas all can enter, but if the person opens an account, they will have their name entered twice.

I am really uneasy about this, but cannot put my finger on why. Everyone will have a chance, just a better chance, if they become a customer.

Anyone been expose to something like this?

I believe you cannot provide an entrant a better chance of winning by their becoming a customer. This would meet the definition of "providing consideration" for a chance to win and would then be a lottery.

I agree with Guitardude. That's why everyone puts "no purchase necessary to win".