We recently introduced a new lower cost Standby state for Live events to enable you to hold a small hotpool of live events in your application that are "ready to go". You can then hand them out to the application, update any settings needed with an Update call and then transition them to "Running" state where they will be ready to receive RTMP ingest data.
I would create the LiveOutput ahead of time, creating the streaming locator before even moving to Running. That way you have the LiveOutput streaming URL ready to go before ingest even begins.
See a similar flow in this Node.js sample (it does not use standby yet...) - https://github.com/Azure-Samples/media-services-v3-node-tutorials/blob/main/AMSv3Samples/Live/index.ts
Sounds like that could help your specific scenario?