Definitely agree that you don't need an MBA to become a Product Manager - the training for Product Management can be quite different to most of what you learn in more traditional Business Administration courses.
That said, I do feel that one can benefit the other - MBA's teach a lot about practically running a business, background learning that can really assist when running a product development and management cycle or team, even if only for the fact that it gives you a better ability to understand and build relationships with other areas of the business.
Really depends on the position a lot of the time, sometimes a Senior Dev role can be very focused on experience over everything else, other times I have seen Senior Dev roles where it was more about mindset and attitude, which might well be the kind of role you're talking about here.
I'd say back yourself and go for it. Always better to dive in the deep end and be challenged than to stay static and not learn
That said, I do feel that one can benefit the other - MBA's teach a lot about practically running a business, background learning that can really assist when running a product development and management cycle or team, even if only for the fact that it gives you a better ability to understand and build relationships with other areas of the business.