Congratulations on the offer!
So how you get into a policymaking career will vary a lot by country. Some countries really value people who have private sector / consulting expertise in policy roles, others value deep subject matter expertise more strongly. I recommend that you look at some of the senior policy makers in your country to see the paths that they took, and even reach out to some who have similar backgrounds to you to talk about how easy it is to get into policy making from the private sector.
It is different from NGOs e.g., UN, EU and even some of the development banks. They tend to place a greater emphasis on having a master's degree and will recruit more from networking where having been exposed to their work e.g., through a consulting project, can be really helpful.
Vis a vis the benefits of an MPA, again this varies by country but can be very helpful for some of the NGOs.
All that being said, consulting gives you a very strong grounding in some of the fundamentals of what it takes to be a great policy maker and if you really want to get into policy making, you'll find a way in. I started my career in government, went to McKinsey, did an MBA, back to McKinsey and now back to government.
If you want to make it work, you will. Take the projects you can find which are government focused, build your network and also keep your eyes open - you may find that you have a passion for adding value in a different industry that you haven't thought about yet.