As with most advice on here, I would recommend that you do not memorise frameworks to simply repeat in an interview. I recently received an offer from McKinsey and also have 10 years experience in government. The general approach I took to government cases was Moore's public value triangle (you can google it). Moore's triangle has three main elements:

1. Public value - what is the public value being created?
2. Legitimacy and support - who will support and authorise the creation of the public value?
3. Operational capabilities - what investments are required to create the public value?

I never used those three elements word-for-word, but instead adapated them to the case. So for example, if I was asked how I might improve traffic conditions for a regional government I might use the following structure:

1. Objectives of traffic reduction - I would want to understand why we want to reduce traffic, who will benefit, whether there are any specific targets etc.
2. Options analysis - do we have specific projects in mind, are the options demand or supply driven
3. Stakeholders interests - I would seek to understand which stakeholders in the community support or oppose the options, what political support exists
4. Implementation - what options exist, what are our capabilities to implement, do we have sufficient funding

You will see that points 1, 3 and 4 essentially match points 1, 2 and 3 (respectively) of the public value triangle. I added in point 2 for completeness. Essentially the public value triangle helped me create a relatively MECE structure to guide the case.

Another very important point with government cases is that you should always clarify whether profit is an objective. For example, in building a road you want to ask whether the government seeks to make a profit from the road, or whether it sole objective is reducing traffic.

Hope that helps!