Bringing justice to international actors opposing democratic ethics is regrettably less of a priority today than enforcing highly publicized and politicized criminal cases.
But to answer your question more fully, you can't solve this problem without supranational cooperation. A "police force" working to safeguard the Internet would have to work under authority of the UN, not any single nation.
It's an association of governments, where they get together and organize things. All the power is in the individual governments. There are some grey areas and exceptions, but overwhelmingly the above is the case.
The UN could coordinate cybercrime law and national agencies.
Based on an estimate of the design of organizations: Sometimes you expand an existing function within an organization, sometimes you add a sub-organzation (e.g., a division), sometimes you create a new organization. Which, when, and why? Standard CEO fare. A couple basic considerations off the top of my head:
Organizations have priorities. As one example, the story (I can't promise perfect details here) is that the US Air Force has always had the priority of pilots - it's run by pilots, they are glorified - strategic bombers and air superiority (air-to-air) fighter planes. Tasked also with providing close air support for ground soldiers, drones for surveillance, and orbital operations, they don't quite get around to those needs: They want bombers and air superiority fighters, flown by pilots, so that's what gets attention, that's what they invest in researching, developing, and buying - F-35's, B-21's, etc. (name a high-price uber-tech platform they've built for close air support, surveillance, or space). For close air support, they insist the F-35 will do it well enough as a secondary function, and want to cut other options - 'well enough' is not the language of priority. It's a constant battle to get them to deliver on these other needs. Partly for that reason, the Marines provide their own air support and the Army has helicopters - they have different priorities than the Air Force - and the US created a separate Space Force.
Organizations also have competencies, which affects the expertise of leaders, the acquired deep organizational knowledge, the asset investments, the organizational structure, and the culture - systems engineers have a different culture than movie actors. If the people in the executive meeting know storage but not networking, you can imagine the results for the networking function. Consider recruiting, training, mentoring, and promotion for networking personnel. Just consider office locations, which will be near the storage talent and facilities, but not near the Internet exchange and networking talent hotbed.
The FBI's priority has been terrorism. Catching domestic terrorists seems much different than investigating cybercrime. The FBI leaders have little expertise in the latter; the entire organization is built around the former. The agent training and skills needed for cybercrime and terrorism seem completely different, the assets needed seem completely different (field offices versus high-performance, highly secure computing centers). I would guess the culture would be very different, with cybercrime placing a very high priority on intellectual ability seated in a room, not interpersonal skill (interviews, etc.), tactical decisions, and physical action around the world. My impression is that a different agency, or at least a major FBI division that reports directly to the top, is needed.