It’s possible to remove your cabinet at a cost of zero political capital. This is not done using the “reshuffle cabinet” button, but it’s end result is effectively the same.
Playing as France, one of my cabinet members support dropped to the point where their background was red - which means I can fire them for zero political capital, which I did. Firing a cabinet member caused the support of all cabinet members to drop slightly, and another went red. Fired them for zero capital. Process continued until entire cabinet was gone, without spending a single political capital point.
I understand that I’ll likely receive some backlash from the public for the firings, but this method allows me to get an extra 10 political points worth of actions than I otherwise would have had.