You can get into policy with just a JD. However, it is easier to get a policy job with a JD and a MPA. A lot of hiring managers are sceptical of hiring JDs into policy roles because they think they are a flight risk or couldn't cut it as a lawyer. Having a JD/MPA helps demonstrate that you aren't a flight risk for policy jobs and actually interested in policy. Similarly, having a JD/MPA makes it easier to demonstrate you actually have an interest in public sector law and didn't just strike out from big law.
I don't know what your financial situation is, but there are some benefits to a JD/MPA.
Nobody on this forum knows you or whether you're the type of person who can pull off the bar exam without studying. Never assume you're that type of person, always go into an exam having studied as much as possible.
There's no shame in deferring. Save yourself some money (and anxiety) and write the exam when you've had the time to adequately prepare.