Note: I initially assumed that you were seeking a refund from your old insurance company, and my original answer is shown first. Below that, I have added a new section to address the clarification that you are seeking a refund from the new insurance company.
Also, please understand that I am not necessarily an expert in health insurance, so my answer below is an educated guess.
The mistake was yours, and what you are asking the old insurance company to do is to refund your insurance premium for a month that has already gone by. If they say they aren’t going to refund that month’s premium, I don’t think there is anything you can do.
Insurance companies don’t generally give refunds after the insured period is over, due to the nature of insurance. You were covered by them, and if you had insurance claims that were greater than the premium, you certainly wouldn’t be seeking a refund.
Another thing to note is that it is possible (and sometimes even desired) to be covered by more than one health insurance company at the same time. So from the insurance company’s perspective, you could have intentionally doubled-up on coverage, and are now asking for a refund after the insured period is over.
I don’t believe they are required to give you a refund at this point, and I don’t believe that there is any way to force them to do so if they are unwilling. However, if you did have any medical claims for that month, you can submit them to both insurance companies (making sure that they both know that you are double covered). They will select one as the “primary” insurance for that month, which will pay normally according to the plan, and the other might cover any remaining out-of-pocket expenses, such as copays or coinsurance. (The name of this process is “Coordination of Benefits” (COB). See this article from netquote.com for an explanation of the process.)
I see with your latest edit that you have clarified that you are seeking a refund from your new insurance company. If I am reading your question correctly, your new insurance company has said that they could do this, but your new benefits provider is saying that you can't. So it appears that you are getting different answers from the two entities. My comments above about insurance companies not being required to give refunds still applies, but if the insurance company is saying that they can do it, perhaps it simply needs to be communicated to the benefits provider. I would go back to whoever you talked to at the insurance company, let them know what the benefits provider said, and see if they can work directly with the benefits provider to get you a refund.
As you talk to people, be nice to everyone, because they don't have to help you out. On the other hand, you have nothing to lose by asking (the worst they can say is "no"), so some insistence might be beneficial.
