The following diagram shows one potential integration between a matchmaker and Arbitrium. If you've integrated your matchmaker with Agones, request an instance from a fleet, like this. Don't worry; the flow remains the same.
Our services only start game components (servers, relays) once needed. There is no need to register, pre-start or warm up your game servers. You can remove this portion of your matchmaker and simplify your rules mechanism. You can also remove the rules around latency and enforce region since Arbitrium will pick the best location based on match context.
You can edit the call-flow using the web sequence website and the following info (cut & paste in the web page).